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}
div.nbgrader_cell .panel-body {
padding: 0.4em;
}
.comment {
width: 100%;
margin-top: 0.5em;
}
div.nbgrader_cell .input_area {
background: white;
border: none;
}
.score {
color: black;
}
li.late-penalty {
color: #d2413a;
}
span.nbgrader-label {
line-height: 25px;
}
.save-icon {
margin-left: 1em;
top: 4px;
}
</style>
<style type="text/css">.MathJax_Hover_Frame {border-radius: .25em; -webkit-border-radius: .25em; -moz-border-radius: .25em; -khtml-border-radius: .25em; box-shadow: 0px 0px 15px #83A; -webkit-box-shadow: 0px 0px 15px #83A; -moz-box-shadow: 0px 0px 15px #83A; -khtml-box-shadow: 0px 0px 15px #83A; border: 1px solid #A6D ! important; display: inline-block; position: absolute}
.MathJax_Menu_Button .MathJax_Hover_Arrow {position: absolute; cursor: pointer; display: inline-block; border: 2px solid #AAA; border-radius: 4px; -webkit-border-radius: 4px; -moz-border-radius: 4px; -khtml-border-radius: 4px; font-family: 'Courier New',Courier; font-size: 9px; color: #F0F0F0}
.MathJax_Menu_Button .MathJax_Hover_Arrow span {display: block; background-color: #AAA; border: 1px solid; border-radius: 3px; line-height: 0; padding: 4px}
.MathJax_Hover_Arrow:hover {color: white!important; border: 2px solid #CCC!important}
.MathJax_Hover_Arrow:hover span {background-color: #CCC!important}
</style><style type="text/css">#MathJax_About {position: fixed; left: 50%; width: auto; text-align: center; border: 3px outset; padding: 1em 2em; background-color: #DDDDDD; color: black; cursor: default; font-family: message-box; font-size: 120%; font-style: normal; text-indent: 0; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; z-index: 201; border-radius: 15px; -webkit-border-radius: 15px; -moz-border-radius: 15px; -khtml-border-radius: 15px; box-shadow: 0px 10px 20px #808080; -webkit-box-shadow: 0px 10px 20px #808080; -moz-box-shadow: 0px 10px 20px #808080; -khtml-box-shadow: 0px 10px 20px #808080; filter: progid:DXImageTransform.Microsoft.dropshadow(OffX=2, OffY=2, Color='gray', Positive='true')}
#MathJax_About.MathJax_MousePost {outline: none}
.MathJax_Menu {position: absolute; background-color: white; color: black; width: auto; padding: 2px; border: 1px solid #CCCCCC; margin: 0; cursor: default; font: menu; text-align: left; text-indent: 0; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; z-index: 201; box-shadow: 0px 10px 20px #808080; -webkit-box-shadow: 0px 10px 20px #808080; -moz-box-shadow: 0px 10px 20px #808080; -khtml-box-shadow: 0px 10px 20px #808080; filter: progid:DXImageTransform.Microsoft.dropshadow(OffX=2, OffY=2, Color='gray', Positive='true')}
.MathJax_MenuItem {padding: 2px 2em; background: transparent}
.MathJax_MenuArrow {position: absolute; right: .5em; padding-top: .25em; color: #666666; font-size: .75em}
.MathJax_MenuActive .MathJax_MenuArrow {color: white}
.MathJax_MenuArrow.RTL {left: .5em; right: auto}
.MathJax_MenuCheck {position: absolute; left: .7em}
.MathJax_MenuCheck.RTL {right: .7em; left: auto}
.MathJax_MenuRadioCheck {position: absolute; left: 1em}
.MathJax_MenuRadioCheck.RTL {right: 1em; left: auto}
.MathJax_MenuLabel {padding: 2px 2em 4px 1.33em; font-style: italic}
.MathJax_MenuRule {border-top: 1px solid #CCCCCC; margin: 4px 1px 0px}
.MathJax_MenuDisabled {color: GrayText}
.MathJax_MenuActive {background-color: Highlight; color: HighlightText}
.MathJax_MenuDisabled:focus, .MathJax_MenuLabel:focus {background-color: #E8E8E8}
.MathJax_ContextMenu:focus {outline: none}
.MathJax_ContextMenu .MathJax_MenuItem:focus {outline: none}
#MathJax_AboutClose {top: .2em; right: .2em}
.MathJax_Menu .MathJax_MenuClose {top: -10px; left: -10px}
.MathJax_MenuClose {position: absolute; cursor: pointer; display: inline-block; border: 2px solid #AAA; border-radius: 18px; -webkit-border-radius: 18px; -moz-border-radius: 18px; -khtml-border-radius: 18px; font-family: 'Courier New',Courier; font-size: 24px; color: #F0F0F0}
.MathJax_MenuClose span {display: block; background-color: #AAA; border: 1.5px solid; border-radius: 18px; -webkit-border-radius: 18px; -moz-border-radius: 18px; -khtml-border-radius: 18px; line-height: 0; padding: 8px 0 6px}
.MathJax_MenuClose:hover {color: white!important; border: 2px solid #CCC!important}
.MathJax_MenuClose:hover span {background-color: #CCC!important}
.MathJax_MenuClose:hover:focus {outline: none}
</style><style type="text/css">.MathJax_Preview .MJXf-math {color: inherit!important}
</style><style type="text/css">.MJX_Assistive_MathML {position: absolute!important; top: 0; left: 0; clip: rect(1px, 1px, 1px, 1px); padding: 1px 0 0 0!important; border: 0!important; height: 1px!important; width: 1px!important; overflow: hidden!important; display: block!important; -webkit-touch-callout: none; -webkit-user-select: none; -khtml-user-select: none; -moz-user-select: none; -ms-user-select: none; user-select: none}
.MJX_Assistive_MathML.MJX_Assistive_MathML_Block {width: 100%!important}
</style><style type="text/css">#MathJax_Zoom {position: absolute; background-color: #F0F0F0; overflow: auto; display: block; z-index: 301; padding: .5em; border: 1px solid black; margin: 0; font-weight: normal; font-style: normal; text-align: left; text-indent: 0; text-transform: none; line-height: normal; letter-spacing: normal; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; -webkit-box-sizing: content-box; -moz-box-sizing: content-box; box-sizing: content-box; box-shadow: 5px 5px 15px #AAAAAA; -webkit-box-shadow: 5px 5px 15px #AAAAAA; -moz-box-shadow: 5px 5px 15px #AAAAAA; -khtml-box-shadow: 5px 5px 15px #AAAAAA; filter: progid:DXImageTransform.Microsoft.dropshadow(OffX=2, OffY=2, Color='gray', Positive='true')}
#MathJax_ZoomOverlay {position: absolute; left: 0; top: 0; z-index: 300; display: inline-block; width: 100%; height: 100%; border: 0; padding: 0; margin: 0; background-color: white; opacity: 0; filter: alpha(opacity=0)}
#MathJax_ZoomFrame {position: relative; display: inline-block; height: 0; width: 0}
#MathJax_ZoomEventTrap {position: absolute; left: 0; top: 0; z-index: 302; display: inline-block; border: 0; padding: 0; margin: 0; background-color: white; opacity: 0; filter: alpha(opacity=0)}
</style><style type="text/css">.MathJax_Preview {color: #888}
#MathJax_Message {position: fixed; left: 1px; bottom: 2px; background-color: #E6E6E6; border: 1px solid #959595; margin: 0px; padding: 2px 8px; z-index: 102; color: black; font-size: 80%; width: auto; white-space: nowrap}
#MathJax_MSIE_Frame {position: absolute; top: 0; left: 0; width: 0px; z-index: 101; border: 0px; margin: 0px; padding: 0px}
.MathJax_Error {color: #CC0000; font-style: italic}
</style><style type="text/css">.MJXp-script {font-size: .8em}
.MJXp-right {-webkit-transform-origin: right; -moz-transform-origin: right; -ms-transform-origin: right; -o-transform-origin: right; transform-origin: right}
.MJXp-bold {font-weight: bold}
.MJXp-italic {font-style: italic}
.MJXp-scr {font-family: MathJax_Script,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-frak {font-family: MathJax_Fraktur,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-sf {font-family: MathJax_SansSerif,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-cal {font-family: MathJax_Caligraphic,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-mono {font-family: MathJax_Typewriter,'Times New Roman',Times,STIXGeneral,serif}
.MJXp-largeop {font-size: 150%}
.MJXp-largeop.MJXp-int {vertical-align: -.2em}
.MJXp-math {display: inline-block; line-height: 1.2; text-indent: 0; font-family: 'Times New Roman',Times,STIXGeneral,serif; white-space: nowrap; border-collapse: collapse}
.MJXp-display {display: block; text-align: center; margin: 1em 0}
.MJXp-math span {display: inline-block}
.MJXp-box {display: block!important; text-align: center}
.MJXp-box:after {content: " "}
.MJXp-rule {display: block!important; margin-top: .1em}
.MJXp-char {display: block!important}
.MJXp-mo {margin: 0 .15em}
.MJXp-mfrac {margin: 0 .125em; vertical-align: .25em}
.MJXp-denom {display: inline-table!important; width: 100%}
.MJXp-denom > * {display: table-row!important}
.MJXp-surd {vertical-align: top}
.MJXp-surd > * {display: block!important}
.MJXp-script-box > * {display: table!important; height: 50%}
.MJXp-script-box > * > * {display: table-cell!important; vertical-align: top}
.MJXp-script-box > *:last-child > * {vertical-align: bottom}
.MJXp-script-box > * > * > * {display: block!important}
.MJXp-mphantom {visibility: hidden}
.MJXp-munderover, .MJXp-munder {display: inline-table!important}
.MJXp-over {display: inline-block!important; text-align: center}
.MJXp-over > * {display: block!important}
.MJXp-munderover > *, .MJXp-munder > * {display: table-row!important}
.MJXp-mtable {vertical-align: .25em; margin: 0 .125em}
.MJXp-mtable > * {display: inline-table!important; vertical-align: middle}
.MJXp-mtr {display: table-row!important}
.MJXp-mtd {display: table-cell!important; text-align: center; padding: .5em 0 0 .5em}
.MJXp-mtr > .MJXp-mtd:first-child {padding-left: 0}
.MJXp-mtr:first-child > .MJXp-mtd {padding-top: 0}
.MJXp-mlabeledtr {display: table-row!important}
.MJXp-mlabeledtr > .MJXp-mtd:first-child {padding-left: 0}
.MJXp-mlabeledtr:first-child > .MJXp-mtd {padding-top: 0}
.MJXp-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 1px 3px; font-style: normal; font-size: 90%}
.MJXp-scale0 {-webkit-transform: scaleX(.0); -moz-transform: scaleX(.0); -ms-transform: scaleX(.0); -o-transform: scaleX(.0); transform: scaleX(.0)}
.MJXp-scale1 {-webkit-transform: scaleX(.1); -moz-transform: scaleX(.1); -ms-transform: scaleX(.1); -o-transform: scaleX(.1); transform: scaleX(.1)}
.MJXp-scale2 {-webkit-transform: scaleX(.2); -moz-transform: scaleX(.2); -ms-transform: scaleX(.2); -o-transform: scaleX(.2); transform: scaleX(.2)}
.MJXp-scale3 {-webkit-transform: scaleX(.3); -moz-transform: scaleX(.3); -ms-transform: scaleX(.3); -o-transform: scaleX(.3); transform: scaleX(.3)}
.MJXp-scale4 {-webkit-transform: scaleX(.4); -moz-transform: scaleX(.4); -ms-transform: scaleX(.4); -o-transform: scaleX(.4); transform: scaleX(.4)}
.MJXp-scale5 {-webkit-transform: scaleX(.5); -moz-transform: scaleX(.5); -ms-transform: scaleX(.5); -o-transform: scaleX(.5); transform: scaleX(.5)}
.MJXp-scale6 {-webkit-transform: scaleX(.6); -moz-transform: scaleX(.6); -ms-transform: scaleX(.6); -o-transform: scaleX(.6); transform: scaleX(.6)}
.MJXp-scale7 {-webkit-transform: scaleX(.7); -moz-transform: scaleX(.7); -ms-transform: scaleX(.7); -o-transform: scaleX(.7); transform: scaleX(.7)}
.MJXp-scale8 {-webkit-transform: scaleX(.8); -moz-transform: scaleX(.8); -ms-transform: scaleX(.8); -o-transform: scaleX(.8); transform: scaleX(.8)}
.MJXp-scale9 {-webkit-transform: scaleX(.9); -moz-transform: scaleX(.9); -ms-transform: scaleX(.9); -o-transform: scaleX(.9); transform: scaleX(.9)}
.MathJax_PHTML .noError {vertical-align: ; font-size: 90%; text-align: left; color: black; padding: 1px 3px; border: 1px solid}
</style><style type="text/css">.mjx-chtml {display: inline-block; line-height: 0; text-indent: 0; text-align: left; text-transform: none; font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; letter-spacing: normal; word-wrap: normal; word-spacing: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0; min-height: 0; border: 0; margin: 0; padding: 1px 0}
.MJXc-display {display: block; text-align: center; margin: 1em 0; padding: 0}
.mjx-chtml[tabindex]:focus, body :focus .mjx-chtml[tabindex] {display: inline-table}
.mjx-full-width {text-align: center; display: table-cell!important; width: 10000em}
.mjx-math {display: inline-block; border-collapse: separate; border-spacing: 0}
.mjx-math * {display: inline-block; -webkit-box-sizing: content-box!important; -moz-box-sizing: content-box!important; box-sizing: content-box!important; text-align: left}
.mjx-numerator {display: block; text-align: center}
.mjx-denominator {display: block; text-align: center}
.MJXc-stacked {height: 0; position: relative}
.MJXc-stacked > * {position: absolute}
.MJXc-bevelled > * {display: inline-block}
.mjx-stack {display: inline-block}
.mjx-op {display: block}
.mjx-under {display: table-cell}
.mjx-over {display: block}
.mjx-over > * {padding-left: 0px!important; padding-right: 0px!important}
.mjx-under > * {padding-left: 0px!important; padding-right: 0px!important}
.mjx-stack > .mjx-sup {display: block}
.mjx-stack > .mjx-sub {display: block}
.mjx-prestack > .mjx-presup {display: block}
.mjx-prestack > .mjx-presub {display: block}
.mjx-delim-h > .mjx-char {display: inline-block}
.mjx-surd {vertical-align: top}
.mjx-mphantom * {visibility: hidden}
.mjx-merror {background-color: #FFFF88; color: #CC0000; border: 1px solid #CC0000; padding: 2px 3px; font-style: normal; font-size: 90%}
.mjx-annotation-xml {line-height: normal}
.mjx-menclose > svg {fill: none; stroke: currentColor}
.mjx-mtr {display: table-row}
.mjx-mlabeledtr {display: table-row}
.mjx-mtd {display: table-cell; text-align: center}
.mjx-label {display: table-row}
.mjx-box {display: inline-block}
.mjx-block {display: block}
.mjx-span {display: inline}
.mjx-char {display: block; white-space: pre}
.mjx-itable {display: inline-table; width: auto}
.mjx-row {display: table-row}
.mjx-cell {display: table-cell}
.mjx-table {display: table; width: 100%}
.mjx-line {display: block; height: 0}
.mjx-strut {width: 0; padding-top: 1em}
.mjx-vsize {width: 0}
.MJXc-space1 {margin-left: .167em}
.MJXc-space2 {margin-left: .222em}
.MJXc-space3 {margin-left: .278em}
.mjx-chartest {display: block; visibility: hidden; position: absolute; top: 0; line-height: normal; font-size: 500%}
.mjx-chartest .mjx-char {display: inline}
.mjx-chartest .mjx-box {padding-top: 1000px}
.MJXc-processing {visibility: hidden; position: fixed; width: 0; height: 0; overflow: hidden}
.MJXc-processed {display: none}
.mjx-test {font-style: normal; font-weight: normal; font-size: 100%; font-size-adjust: none; text-indent: 0; text-transform: none; letter-spacing: normal; word-spacing: normal; overflow: hidden; height: 1px}
.mjx-test.mjx-test-display {display: table!important}
.mjx-test.mjx-test-inline {display: inline!important; margin-right: -1px}
.mjx-test.mjx-test-default {display: block!important; clear: both}
.mjx-ex-box {display: inline-block!important; position: absolute; overflow: hidden; min-height: 0; max-height: none; padding: 0; border: 0; margin: 0; width: 1px; height: 60ex}
.mjx-test-inline .mjx-left-box {display: inline-block; width: 0; float: left}
.mjx-test-inline .mjx-right-box {display: inline-block; width: 0; float: right}
.mjx-test-display .mjx-right-box {display: table-cell!important; width: 10000em!important; min-width: 0; max-width: none; padding: 0; border: 0; margin: 0}
#MathJax_CHTML_Tooltip {background-color: InfoBackground; color: InfoText; border: 1px solid black; box-shadow: 2px 2px 5px #AAAAAA; -webkit-box-shadow: 2px 2px 5px #AAAAAA; -moz-box-shadow: 2px 2px 5px #AAAAAA; -khtml-box-shadow: 2px 2px 5px #AAAAAA; padding: 3px 4px; z-index: 401; position: absolute; left: 0; top: 0; width: auto; height: auto; display: none}
.mjx-chtml .mjx-noError {line-height: 1.2; vertical-align: ; font-size: 90%; text-align: left; color: black; padding: 1px 3px; border: 1px solid}
.MJXc-TeX-unknown-R {font-family: STIXGeneral,'Cambria Math','Arial Unicode MS',serif; font-style: normal; font-weight: normal}
.MJXc-TeX-unknown-I {font-family: STIXGeneral,'Cambria Math','Arial Unicode MS',serif; font-style: italic; font-weight: normal}
.MJXc-TeX-unknown-B {font-family: STIXGeneral,'Cambria Math','Arial Unicode MS',serif; font-style: normal; font-weight: bold}
.MJXc-TeX-unknown-BI {font-family: STIXGeneral,'Cambria Math','Arial Unicode MS',serif; font-style: italic; font-weight: bold}
.MJXc-TeX-ams-R {font-family: MJXc-TeX-ams-R,MJXc-TeX-ams-Rw}
.MJXc-TeX-cal-B {font-family: MJXc-TeX-cal-B,MJXc-TeX-cal-Bx,MJXc-TeX-cal-Bw}
.MJXc-TeX-frak-R {font-family: MJXc-TeX-frak-R,MJXc-TeX-frak-Rw}
.MJXc-TeX-frak-B {font-family: MJXc-TeX-frak-B,MJXc-TeX-frak-Bx,MJXc-TeX-frak-Bw}
.MJXc-TeX-math-BI {font-family: MJXc-TeX-math-BI,MJXc-TeX-math-BIx,MJXc-TeX-math-BIw}
.MJXc-TeX-sans-R {font-family: MJXc-TeX-sans-R,MJXc-TeX-sans-Rw}
.MJXc-TeX-sans-B {font-family: MJXc-TeX-sans-B,MJXc-TeX-sans-Bx,MJXc-TeX-sans-Bw}
.MJXc-TeX-sans-I {font-family: MJXc-TeX-sans-I,MJXc-TeX-sans-Ix,MJXc-TeX-sans-Iw}
.MJXc-TeX-script-R {font-family: MJXc-TeX-script-R,MJXc-TeX-script-Rw}
.MJXc-TeX-type-R {font-family: MJXc-TeX-type-R,MJXc-TeX-type-Rw}
.MJXc-TeX-cal-R {font-family: MJXc-TeX-cal-R,MJXc-TeX-cal-Rw}
.MJXc-TeX-main-B {font-family: MJXc-TeX-main-B,MJXc-TeX-main-Bx,MJXc-TeX-main-Bw}
.MJXc-TeX-main-I {font-family: MJXc-TeX-main-I,MJXc-TeX-main-Ix,MJXc-TeX-main-Iw}
.MJXc-TeX-main-R {font-family: MJXc-TeX-main-R,MJXc-TeX-main-Rw}
.MJXc-TeX-math-I {font-family: MJXc-TeX-math-I,MJXc-TeX-math-Ix,MJXc-TeX-math-Iw}
.MJXc-TeX-size1-R {font-family: MJXc-TeX-size1-R,MJXc-TeX-size1-Rw}
.MJXc-TeX-size2-R {font-family: MJXc-TeX-size2-R,MJXc-TeX-size2-Rw}
.MJXc-TeX-size3-R {font-family: MJXc-TeX-size3-R,MJXc-TeX-size3-Rw}
.MJXc-TeX-size4-R {font-family: MJXc-TeX-size4-R,MJXc-TeX-size4-Rw}
.MJXc-TeX-vec-R {font-family: MJXc-TeX-vec-R,MJXc-TeX-vec-Rw}
.MJXc-TeX-vec-B {font-family: MJXc-TeX-vec-B,MJXc-TeX-vec-Bx,MJXc-TeX-vec-Bw}
@font-face {font-family: MJXc-TeX-ams-R; src: local('MathJax_AMS'), local('MathJax_AMS-Regular')}
@font-face {font-family: MJXc-TeX-ams-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/fonts/HTML-CSS/TeX/eot/MathJax_AMS-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/fonts/HTML-CSS/TeX/woff/MathJax_AMS-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/fonts/HTML-CSS/TeX/otf/MathJax_AMS-Regular.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-cal-B; src: local('MathJax_Caligraphic Bold'), local('MathJax_Caligraphic-Bold')}
@font-face {font-family: MJXc-TeX-cal-Bx; src: local('MathJax_Caligraphic'); font-weight: bold}
@font-face {font-family: MJXc-TeX-cal-Bw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/fonts/HTML-CSS/TeX/eot/MathJax_Caligraphic-Bold.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/fonts/HTML-CSS/TeX/woff/MathJax_Caligraphic-Bold.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/fonts/HTML-CSS/TeX/otf/MathJax_Caligraphic-Bold.otf') format('opentype')}
@font-face {font-family: MJXc-TeX-frak-R; src: local('MathJax_Fraktur'), local('MathJax_Fraktur-Regular')}
@font-face {font-family: MJXc-TeX-frak-Rw; src /*1*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/fonts/HTML-CSS/TeX/eot/MathJax_Fraktur-Regular.eot'); src /*2*/: url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/fonts/HTML-CSS/TeX/woff/MathJax_Fraktur-Regular.woff') format('woff'), url('https://cdnjs.cloudflare.com/ajax/libs/mathjax/2.7.7/fonts/HTML-CSS/TeX/otf/MathJax_Fraktur-Regular.otf') format('opentype')}
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<h4>exercise_sheet_01 (Score: 97.0 / 100.0)</h4>
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<h1 id="Exercise-sheet">Exercise sheet<a class="anchor-link" href="#Exercise-sheet"></a></h1><p>Some general remarks about the exercises:</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">NAME</span> <span class="o">=</span> <span class="s2">"Kees van Kempen"</span>
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<p><strong>Exercise sheet 1</strong></p>
<p>Code from the lecture:</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">numpy</span> <span class="k">as</span> <span class="nn">np</span>
<span class="kn">import</span> <span class="nn">matplotlib.pylab</span> <span class="k">as</span> <span class="nn">plt</span>
<span class="n">rng</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">random</span><span class="o">.</span><span class="n">default_rng</span><span class="p">()</span>
<span class="o">%</span><span class="k">matplotlib</span> inline
<span class="k">def</span> <span class="nf">random_in_square</span><span class="p">():</span>
<span class="sd">"""Returns a random position in the square [-1,1)x[-1,1)."""</span>
<span class="k">return</span> <span class="n">rng</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="mi">2</span><span class="p">)</span>
<span class="k">def</span> <span class="nf">is_in_circle</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">dot</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">x</span><span class="p">)</span> <span class="o">&lt;</span> <span class="mi">1</span>
<span class="k">def</span> <span class="nf">simulate_number_of_hits</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="sd">"""Simulates number of hits in case of N trials in the pebble game."""</span>
<span class="n">number_hits</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">position</span> <span class="o">=</span> <span class="n">random_in_square</span><span class="p">()</span>
<span class="k">if</span> <span class="n">is_in_circle</span><span class="p">(</span><span class="n">position</span><span class="p">):</span>
<span class="n">number_hits</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">number_hits</span>
<span class="k">def</span> <span class="nf">random_in_disk</span><span class="p">():</span>
<span class="sd">"""Returns a uniform point in the unit disk via rejection."""</span>
<span class="n">position</span> <span class="o">=</span> <span class="n">random_in_square</span><span class="p">()</span>
<span class="k">while</span> <span class="ow">not</span> <span class="n">is_in_circle</span><span class="p">(</span><span class="n">position</span><span class="p">):</span>
<span class="n">position</span> <span class="o">=</span> <span class="n">random_in_square</span><span class="p">()</span>
<span class="k">return</span> <span class="n">position</span>
<span class="k">def</span> <span class="nf">is_in_square</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
<span class="sd">"""Returns True if x is in the square (-1,1)^2."""</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">0</span><span class="p">])</span> <span class="o">&lt;</span> <span class="mi">1</span> <span class="ow">and</span> <span class="n">np</span><span class="o">.</span><span class="n">abs</span><span class="p">(</span><span class="n">x</span><span class="p">[</span><span class="mi">1</span><span class="p">])</span> <span class="o">&lt;</span> <span class="mi">1</span>
<span class="k">def</span> <span class="nf">sample_next_position_naively</span><span class="p">(</span><span class="n">position</span><span class="p">,</span><span class="n">delta</span><span class="p">):</span>
<span class="sd">"""Keep trying a throw until it ends up in the square."""</span>
<span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
<span class="n">next_position</span> <span class="o">=</span> <span class="n">position</span> <span class="o">+</span> <span class="n">delta</span><span class="o">*</span><span class="n">random_in_disk</span><span class="p">()</span>
<span class="k">if</span> <span class="n">is_in_square</span><span class="p">(</span><span class="n">next_position</span><span class="p">):</span>
<span class="k">return</span> <span class="n">next_position</span>
<span class="k">def</span> <span class="nf">naive_markov_pebble</span><span class="p">(</span><span class="n">start</span><span class="p">,</span><span class="n">delta</span><span class="p">,</span><span class="n">N</span><span class="p">):</span>
<span class="sd">"""Simulates the number of hits in the naive Markov-chain version </span>
<span class="sd"> of the pebble game."""</span>
<span class="n">number_hits</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">position</span> <span class="o">=</span> <span class="n">start</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">position</span> <span class="o">=</span> <span class="n">sample_next_position_naively</span><span class="p">(</span><span class="n">position</span><span class="p">,</span><span class="n">delta</span><span class="p">)</span>
<span class="k">if</span> <span class="n">is_in_circle</span><span class="p">(</span><span class="n">position</span><span class="p">):</span>
<span class="n">number_hits</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">number_hits</span>
<span class="k">def</span> <span class="nf">naive_markov_pebble_generator</span><span class="p">(</span><span class="n">start</span><span class="p">,</span><span class="n">delta</span><span class="p">,</span><span class="n">N</span><span class="p">):</span>
<span class="sd">"""Same as naive_markov_pebble but only yields the positions."""</span>
<span class="n">position</span> <span class="o">=</span> <span class="n">start</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">position</span> <span class="o">=</span> <span class="n">sample_next_position_naively</span><span class="p">(</span><span class="n">position</span><span class="p">,</span><span class="n">delta</span><span class="p">)</span>
<span class="k">yield</span> <span class="n">position</span>
<span class="k">def</span> <span class="nf">sample_next_position</span><span class="p">(</span><span class="n">position</span><span class="p">,</span><span class="n">delta</span><span class="p">):</span>
<span class="sd">"""Attempt a throw and reject when outside the square."""</span>
<span class="n">next_position</span> <span class="o">=</span> <span class="n">position</span> <span class="o">+</span> <span class="n">delta</span><span class="o">*</span><span class="n">random_in_disk</span><span class="p">()</span>
<span class="k">if</span> <span class="n">is_in_square</span><span class="p">(</span><span class="n">next_position</span><span class="p">):</span>
<span class="k">return</span> <span class="n">next_position</span> <span class="c1"># accept!</span>
<span class="k">else</span><span class="p">:</span>
<span class="k">return</span> <span class="n">position</span> <span class="c1"># reject!</span>
<span class="k">def</span> <span class="nf">markov_pebble</span><span class="p">(</span><span class="n">start</span><span class="p">,</span><span class="n">delta</span><span class="p">,</span><span class="n">N</span><span class="p">):</span>
<span class="sd">"""Simulates the number of hits in the proper Markov-chain version of the pebble game."""</span>
<span class="n">number_hits</span> <span class="o">=</span> <span class="mi">0</span>
<span class="n">position</span> <span class="o">=</span> <span class="n">start</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">position</span> <span class="o">=</span> <span class="n">sample_next_position</span><span class="p">(</span><span class="n">position</span><span class="p">,</span><span class="n">delta</span><span class="p">)</span>
<span class="k">if</span> <span class="n">is_in_circle</span><span class="p">(</span><span class="n">position</span><span class="p">):</span>
<span class="n">number_hits</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">number_hits</span>
<span class="k">def</span> <span class="nf">markov_pebble_generator</span><span class="p">(</span><span class="n">start</span><span class="p">,</span><span class="n">delta</span><span class="p">,</span><span class="n">N</span><span class="p">):</span>
<span class="sd">"""Same as markov_pebble but only yields the positions."""</span>
<span class="n">position</span> <span class="o">=</span> <span class="n">start</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">position</span> <span class="o">=</span> <span class="n">sample_next_position</span><span class="p">(</span><span class="n">position</span><span class="p">,</span><span class="n">delta</span><span class="p">)</span>
<span class="k">yield</span> <span class="n">position</span>
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<h2 id="Empirical-convergence-rate-in-the-pebble-game">Empirical convergence rate in the pebble game<a class="anchor-link" href="#Empirical-convergence-rate-in-the-pebble-game"></a></h2><p><strong>(a)</strong> Write a function <code>pi_stddev</code> that estimates the standard deviation <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-1-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-1" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-2" class="mjx-mrow"><span id="MJXc-Node-3" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.186em; padding-bottom: 0.308em; padding-right: 0.001em;">σ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi></math></span></span><script type="math/tex" id="MathJax-Element-1">\sigma</script> of the <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-2-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-4" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-5" class="mjx-mrow"><span id="MJXc-Node-6" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.186em; padding-bottom: 0.308em; padding-right: 0.003em;">π</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span></span><script type="math/tex" id="MathJax-Element-2">\pi</script>-estimate using <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-3-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-7" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-8" class="mjx-mrow"><span id="MJXc-Node-9" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.308em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-3">n</script> trials by running the direct-sampling pebble game <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-4-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-10" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-11" class="mjx-mrow"><span id="MJXc-Node-12" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.308em;">m</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi></math></span></span><script type="math/tex" id="MathJax-Element-4">m</script> times. Store this data for <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-5-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mn&gt;5&lt;/mn&gt;&lt;/msup&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;msup&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mn&gt;14&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-13" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-14" class="mjx-mrow"><span id="MJXc-Node-15" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.308em;">n</span></span><span id="MJXc-Node-16" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.064em; padding-bottom: 0.308em;">=</span></span><span id="MJXc-Node-17" class="mjx-msubsup MJXc-space3"><span class="mjx-base"><span id="MJXc-Node-18" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-19" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">4</span></span></span></span><span id="MJXc-Node-20" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.18em; padding-bottom: 0.553em;">,</span></span><span id="MJXc-Node-21" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-22" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-23" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">5</span></span></span></span><span id="MJXc-Node-24" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.18em; padding-bottom: 0.553em;">,</span></span><span id="MJXc-Node-25" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.18em; padding-bottom: 0.369em;">…</span></span><span id="MJXc-Node-26" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.18em; padding-bottom: 0.553em;">,</span></span><span id="MJXc-Node-27" class="mjx-msubsup MJXc-space1"><span class="mjx-base"><span id="MJXc-Node-28" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">2</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.591em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-29" class="mjx-texatom" style=""><span id="MJXc-Node-30" class="mjx-mrow"><span id="MJXc-Node-31" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">14</span></span></span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>n</mi><mo>=</mo><msup><mn>2</mn><mn>4</mn></msup><mo>,</mo><msup><mn>2</mn><mn>5</mn></msup><mo>,</mo><mo>…</mo><mo>,</mo><msup><mn>2</mn><mrow class="MJX-TeXAtom-ORD"><mn>14</mn></mrow></msup></math></span></span><script type="math/tex" id="MathJax-Element-5">n=2^4,2^5,\ldots,2^{14}</script> and <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-6-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;m&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;200&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-32" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-33" class="mjx-mrow"><span id="MJXc-Node-34" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.308em;">m</span></span><span id="MJXc-Node-35" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.064em; padding-bottom: 0.308em;">=</span></span><span id="MJXc-Node-36" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">200</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>m</mi><mo>=</mo><mn>200</mn></math></span></span><script type="math/tex" id="MathJax-Element-6">m=200</script> in an array. <em>Hint</em>: you may use the NumPy function <a href="https://numpy.org/doc/stable/reference/generated/numpy.std.html"><code>np.std</code></a>. <strong>(12 pts)</strong></p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">pi_stddev</span><span class="p">(</span><span class="n">n</span><span class="p">,</span><span class="n">m</span><span class="p">):</span>
<span class="sd">"""Estimate the standard deviation in the pi estimate in the case of n trials,</span>
<span class="sd"> based on m runs of the direct-sampling pebble game."""</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">([</span><span class="n">simulate_number_of_hits</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="n">n</span><span class="o">*</span><span class="mi">4</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">)])</span>
<span class="n">stddev_data</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mi">2</span><span class="o">**</span><span class="n">k</span><span class="p">,</span><span class="n">pi_stddev</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="n">k</span><span class="p">,</span><span class="mi">200</span><span class="p">)]</span> <span class="k">for</span> <span class="n">k</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">4</span><span class="p">,</span><span class="mi">12</span><span class="p">)])</span>
<span class="n">stddev_data</span>
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<pre>array([[1.60000000e+01, 3.54522126e-01],
[3.20000000e+01, 2.87754643e-01],
[6.40000000e+01, 2.21671341e-01],
[1.28000000e+02, 1.50942352e-01],
[2.56000000e+02, 1.00401902e-01],
[5.12000000e+02, 6.94667450e-02],
[1.02400000e+03, 4.96484068e-02],
[2.04800000e+03, 3.59752212e-02]])</pre>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># If done correctly, your code should pass the following tests</span>
<span class="kn">from</span> <span class="nn">nose.tools</span> <span class="kn">import</span> <span class="n">assert_almost_equal</span>
<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">pi_stddev</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mi">3</span><span class="p">,</span><span class="mi">1000</span><span class="p">),</span><span class="mf">0.582</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.03</span><span class="p">)</span>
<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">pi_stddev</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mi">4</span><span class="p">,</span><span class="mi">1000</span><span class="p">),</span><span class="mf">0.411</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.03</span><span class="p">)</span>
<span class="c1">### BEGIN HIDDEN TESTS</span>
<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">pi_stddev</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mi">5</span><span class="p">,</span><span class="mi">1000</span><span class="p">),</span><span class="mf">0.291</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.03</span><span class="p">)</span>
<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">pi_stddev</span><span class="p">(</span><span class="mi">2</span><span class="o">**</span><span class="mi">6</span><span class="p">,</span><span class="mi">1000</span><span class="p">),</span><span class="mf">0.207</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.03</span><span class="p">)</span>
<span class="c1">### END HIDDEN TESTS</span>
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<p><strong>(b)</strong> Write a function <code>fit_power_law</code> that takes an array of <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-7-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-37" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-38" class="mjx-mrow"><span id="MJXc-Node-39" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.491em; padding-bottom: 0.614em;">(</span></span><span id="MJXc-Node-40" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.308em;">n</span></span><span id="MJXc-Node-41" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.18em; padding-bottom: 0.553em;">,</span></span><span id="MJXc-Node-42" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.186em; padding-bottom: 0.308em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-43" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.491em; padding-bottom: 0.614em;">)</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo stretchy="false">)</mo></math></span></span><script type="math/tex" id="MathJax-Element-7">(n,\sigma)</script> pairs and determines best-fit parameters <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-8-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-44" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-45" class="mjx-mrow"><span id="MJXc-Node-46" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.308em;">a</span></span><span id="MJXc-Node-47" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.18em; padding-bottom: 0.553em;">,</span></span><span id="MJXc-Node-48" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.491em;">p</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>,</mo><mi>p</mi></math></span></span><script type="math/tex" id="MathJax-Element-8">a,p</script> for the curve <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-9-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-49" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-50" class="mjx-mrow"><span id="MJXc-Node-51" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.186em; padding-bottom: 0.308em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-52" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.064em; padding-bottom: 0.308em;">=</span></span><span id="MJXc-Node-53" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.308em;">a</span></span><span id="MJXc-Node-54" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-55" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.308em;">n</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-56" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.491em;">p</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mo>=</mo><mi>a</mi><msup><mi>n</mi><mi>p</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-9">\sigma = a n^p</script>. This is best done by fitting a straight line to the data on a log-log scale (<span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-10-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;mi&gt;log&lt;/mi&gt;&lt;mo&gt;&amp;#x2061;&lt;/mo&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-57" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-58" class="mjx-mrow"><span id="MJXc-Node-59" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.43em; padding-bottom: 0.553em;">log</span></span><span id="MJXc-Node-60" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-61" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.186em; padding-bottom: 0.308em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-62" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.064em; padding-bottom: 0.308em;">=</span></span><span id="MJXc-Node-63" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.43em; padding-bottom: 0.553em;">log</span></span><span id="MJXc-Node-64" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-65" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.308em;">a</span></span><span id="MJXc-Node-66" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.308em; padding-bottom: 0.43em;">+</span></span><span id="MJXc-Node-67" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.491em;">p</span></span><span id="MJXc-Node-68" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.43em; padding-bottom: 0.553em;">log</span></span><span id="MJXc-Node-69" class="mjx-mo"><span class="mjx-char"></span></span><span id="MJXc-Node-70" class="mjx-mi MJXc-space1"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.308em;">n</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>log</mi><mo>⁡</mo><mi>σ</mi><mo>=</mo><mi>log</mi><mo>⁡</mo><mi>a</mi><mo>+</mo><mi>p</mi><mi>log</mi><mo>⁡</mo><mi>n</mi></math></span></span><script type="math/tex" id="MathJax-Element-10">\log\sigma=\log a+p\log n</script>). <em>Hint</em>: use <a href="https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.curve_fit.html"><code>curve_fit</code></a> from SciPy. <strong>(12 pts)</strong></p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">scipy.optimize</span>
<span class="k">def</span> <span class="nf">fit_power_law</span><span class="p">(</span><span class="n">stddev_data</span><span class="p">):</span>
<span class="sd">"""Compute the best fit parameters a and p."""</span>
<span class="c1"># Define the to be fitted function. We use the log, due to the calculation of the uncertainties.</span>
<span class="n">log_f</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">n</span><span class="p">,</span> <span class="n">a</span><span class="p">,</span> <span class="n">p</span><span class="p">:</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">a</span><span class="p">)</span> <span class="o">+</span> <span class="n">p</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="n">n</span><span class="p">,</span> <span class="n">𝜎</span> <span class="o">=</span> <span class="n">stddev_data</span><span class="o">.</span><span class="n">T</span>
<span class="n">a_fit</span><span class="p">,</span> <span class="n">p_fit</span> <span class="o">=</span> <span class="n">scipy</span><span class="o">.</span><span class="n">optimize</span><span class="o">.</span><span class="n">curve_fit</span><span class="p">(</span><span class="n">log_f</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">log</span><span class="p">(</span><span class="n">𝜎</span><span class="p">))[</span><span class="mi">0</span><span class="p">]</span>
<span class="k">return</span> <span class="n">a_fit</span><span class="p">,</span> <span class="n">p_fit</span>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">nose.tools</span> <span class="kn">import</span> <span class="n">assert_almost_equal</span>
<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">fit_power_law</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">16.0</span><span class="p">,</span><span class="mf">1.4</span><span class="p">],[</span><span class="mf">32.0</span><span class="p">,</span><span class="mf">1.1</span><span class="p">],[</span><span class="mf">64.0</span><span class="p">,</span><span class="mf">0.9</span><span class="p">]]))[</span><span class="mi">0</span><span class="p">],</span><span class="mf">3.36</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.05</span><span class="p">)</span>
<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">fit_power_law</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([[</span><span class="mf">16.0</span><span class="p">,</span><span class="mf">1.4</span><span class="p">],[</span><span class="mf">32.0</span><span class="p">,</span><span class="mf">1.1</span><span class="p">],[</span><span class="mf">64.0</span><span class="p">,</span><span class="mf">0.9</span><span class="p">]]))[</span><span class="mi">1</span><span class="p">],</span><span class="o">-</span><span class="mf">0.31</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.03</span><span class="p">)</span>
<span class="c1">### BEGIN HIDDEN TESTS</span>
<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">fit_power_law</span><span class="p">(</span><span class="n">stddev_data</span><span class="p">)[</span><span class="mi">0</span><span class="p">],</span><span class="mf">1.82</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">fit_power_law</span><span class="p">(</span><span class="n">stddev_data</span><span class="p">)[</span><span class="mi">1</span><span class="p">],</span><span class="o">-</span><span class="mf">0.5</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.08</span><span class="p">)</span>
<span class="c1">### END HIDDEN TESTS</span>
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<p><strong>(c)</strong> Plot the data against the fitted curve <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-11-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03C3;&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mi&gt;a&lt;/mi&gt;&lt;msup&gt;&lt;mi&gt;n&lt;/mi&gt;&lt;mi&gt;p&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-71" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-72" class="mjx-mrow"><span id="MJXc-Node-73" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.186em; padding-bottom: 0.308em; padding-right: 0.001em;">σ</span></span><span id="MJXc-Node-74" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.064em; padding-bottom: 0.308em;">=</span></span><span id="MJXc-Node-75" class="mjx-mi MJXc-space3"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.308em;">a</span></span><span id="MJXc-Node-76" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-77" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.308em;">n</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-78" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.247em; padding-bottom: 0.491em;">p</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>σ</mi><mo>=</mo><mi>a</mi><msup><mi>n</mi><mi>p</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-11">\sigma = a n^p</script> in a log-log plot. Don't forget to properly label your axes. <em>Hint</em>: use <a href="https://matplotlib.org/stable/api/_as_gen/matplotlib.pyplot.loglog.html"><code>loglog</code></a> from matplotlib. <strong>(12 pts)</strong></p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">a_fit</span><span class="p">,</span> <span class="n">p_fit</span> <span class="o">=</span> <span class="n">fit_power_law</span><span class="p">(</span><span class="n">stddev_data</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Convergence of standard deviation of direct-sampled $\pi$ guesses over $n$ throws among $m = 200$ trials"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">loglog</span><span class="p">(</span><span class="o">*</span><span class="n">stddev_data</span><span class="o">.</span><span class="n">T</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"data"</span><span class="p">)</span>
<span class="n">n_range</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">logspace</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">loglog</span><span class="p">(</span><span class="n">n_range</span><span class="p">,</span> <span class="n">a_fit</span><span class="o">*</span><span class="n">n_range</span><span class="o">**</span><span class="n">p_fit</span><span class="p">,</span>
<span class="n">label</span><span class="o">=</span><span class="s2">"fitted $\sigma = an^p$</span><span class="se">\n</span><span class="s2">($a = </span><span class="si">{:.3f}</span><span class="s2">, p = </span><span class="si">{:.3f}</span><span class="s2">)$"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">a_fit</span><span class="p">,</span> <span class="n">p_fit</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"$n$"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"$\sigma$"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<b>Comments:</b> <p>Good labeling! Use dots for the data.</p>
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<h2 id="Volume-of-a-unit-ball-in-other-dimensions">Volume of a unit ball in other dimensions<a class="anchor-link" href="#Volume-of-a-unit-ball-in-other-dimensions"></a></h2><p>In this exercise you will adapt the direct-sampling Monte Carlo method of the pebble game to higher dimensions to
estimate the <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-12-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-79" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-80" class="mjx-mrow"><span id="MJXc-Node-81" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.003em;">d</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span><script type="math/tex" id="MathJax-Element-12">d</script>-dimensional volume of the <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-13-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-82" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-83" class="mjx-mrow"><span id="MJXc-Node-84" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.003em;">d</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span><script type="math/tex" id="MathJax-Element-13">d</script>-dimensional ball of radius <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-14-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-85" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-86" class="mjx-mrow"><span id="MJXc-Node-87" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">1</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn></math></span></span><script type="math/tex" id="MathJax-Element-14">1</script>.</p>
<p><strong>(a)</strong> Adapt the direct-sampling Monte Carlo method to the pebble game in a <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-15-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-88" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-89" class="mjx-mrow"><span id="MJXc-Node-90" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.003em;">d</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span><script type="math/tex" id="MathJax-Element-15">d</script>-dimensional hypercube <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-16-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/msup&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-91" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-92" class="mjx-mrow"><span id="MJXc-Node-93" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.491em; padding-bottom: 0.614em;">(</span></span><span id="MJXc-Node-94" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.308em; padding-bottom: 0.43em;">−</span></span><span id="MJXc-Node-95" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">1</span></span><span id="MJXc-Node-96" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.18em; padding-bottom: 0.553em;">,</span></span><span id="MJXc-Node-97" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">1</span></span><span id="MJXc-Node-98" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-99" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.491em; padding-bottom: 0.614em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-100" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.003em;">d</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo>,</mo><mn>1</mn><msup><mo stretchy="false">)</mo><mi>d</mi></msup></math></span></span><script type="math/tex" id="MathJax-Element-16">(-1,1)^d</script> with an inscribed <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-17-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-101" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-102" class="mjx-mrow"><span id="MJXc-Node-103" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.003em;">d</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span><script type="math/tex" id="MathJax-Element-17">d</script>-dimensional unit ball. <strong>(12 pts)</strong></p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">number_of_hits_in_d_dimensions</span><span class="p">(</span><span class="n">N</span><span class="p">,</span><span class="n">d</span><span class="p">):</span>
<span class="sd">"""Simulates number of hits in case of N trials in the d-dimensional direct-sampling pebble game."""</span>
<span class="k">def</span> <span class="nf">random_in_d_dimensional_box</span><span class="p">():</span>
<span class="sd">"""Returns a random position in the box [-1,1)^d."""</span>
<span class="k">return</span> <span class="n">rng</span><span class="o">.</span><span class="n">uniform</span><span class="p">(</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span><span class="mi">1</span><span class="p">,</span><span class="n">d</span><span class="p">)</span>
<span class="n">number_hits</span> <span class="o">=</span> <span class="mi">0</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">N</span><span class="p">):</span>
<span class="n">position</span> <span class="o">=</span> <span class="n">random_in_d_dimensional_box</span><span class="p">()</span>
<span class="k">if</span> <span class="n">is_in_circle</span><span class="p">(</span><span class="n">position</span><span class="p">):</span>
<span class="n">number_hits</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">return</span> <span class="n">number_hits</span>
</pre></div>
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<a name="cell-e8e97677ad30041e"></a><div class="panel-heading"><span class="nbgrader-label">Grade cell: <code>cell-e8e97677ad30041e</code></span>
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Score: 12.0 / 12.0 <a href="#top">(Top)</a>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">from</span> <span class="nn">nose.tools</span> <span class="kn">import</span> <span class="n">assert_almost_equal</span>
<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">number_of_hits_in_d_dimensions</span><span class="p">(</span><span class="mi">100</span><span class="p">,</span><span class="mi">1</span><span class="p">),</span><span class="mi">100</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mi">1</span><span class="p">)</span>
<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">number_of_hits_in_d_dimensions</span><span class="p">(</span><span class="mi">2000</span><span class="p">,</span><span class="mi">3</span><span class="p">),</span><span class="mi">1045</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
<span class="c1">### BEGIN HIDDEN TESTS</span>
<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">number_of_hits_in_d_dimensions</span><span class="p">(</span><span class="mi">2000</span><span class="p">,</span><span class="mi">2</span><span class="p">),</span><span class="mi">1572</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">number_of_hits_in_d_dimensions</span><span class="p">(</span><span class="mi">2000</span><span class="p">,</span><span class="mi">4</span><span class="p">),</span><span class="mi">619</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mi">50</span><span class="p">)</span>
<span class="c1">### END HIDDEN TESTS</span>
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<p><strong>(b)</strong> Compare your estimates for the volume <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-18-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/msub&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-104" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-105" class="mjx-mrow"><span id="MJXc-Node-106" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.186em;"><span id="MJXc-Node-107" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.186em;">V</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-108" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.003em;">d</span></span></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>d</mi></msub></math></span></span><script type="math/tex" id="MathJax-Element-18">V_d</script> of the <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-19-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-109" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-110" class="mjx-mrow"><span id="MJXc-Node-111" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.003em;">d</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi></math></span></span><script type="math/tex" id="MathJax-Element-19">d</script>-dimensional unit ball for <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-20-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;N&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;10000&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-112" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-113" class="mjx-mrow"><span id="MJXc-Node-114" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.085em;">N</span></span><span id="MJXc-Node-115" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.064em; padding-bottom: 0.308em;">=</span></span><span id="MJXc-Node-116" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">10000</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>N</mi><mo>=</mo><mn>10000</mn></math></span></span><script type="math/tex" id="MathJax-Element-20">N=10000</script> trials and <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-21-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mo&gt;&amp;#x2026;&lt;/mo&gt;&lt;mo&gt;,&lt;/mo&gt;&lt;mn&gt;7&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-117" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-118" class="mjx-mrow"><span id="MJXc-Node-119" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-120" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.064em; padding-bottom: 0.308em;">=</span></span><span id="MJXc-Node-121" class="mjx-mn MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">1</span></span><span id="MJXc-Node-122" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.18em; padding-bottom: 0.553em;">,</span></span><span id="MJXc-Node-123" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">2</span></span><span id="MJXc-Node-124" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.18em; padding-bottom: 0.553em;">,</span></span><span id="MJXc-Node-125" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.18em; padding-bottom: 0.369em;">…</span></span><span id="MJXc-Node-126" class="mjx-mo MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="margin-top: -0.18em; padding-bottom: 0.553em;">,</span></span><span id="MJXc-Node-127" class="mjx-mn MJXc-space1"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">7</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>d</mi><mo>=</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mn>7</mn></math></span></span><script type="math/tex" id="MathJax-Element-21">d=1,2,\ldots,7</script> to the <a href="https://en.wikipedia.org/wiki/Volume_of_an_n-ball">exact formula</a> <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-22-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;msub&gt;&lt;mi&gt;V&lt;/mi&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;/msub&gt;&lt;mo&gt;=&lt;/mo&gt;&lt;mfrac&gt;&lt;msup&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mrow&gt;&lt;/msup&gt;&lt;mrow&gt;&lt;mi mathvariant=&quot;normal&quot;&gt;&amp;#x0393;&lt;/mi&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mstyle displaystyle=&quot;false&quot; scriptlevel=&quot;0&quot;&gt;&lt;mfrac&gt;&lt;mi&gt;d&lt;/mi&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/mfrac&gt;&lt;/mstyle&gt;&lt;mo&gt;+&lt;/mo&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;/mrow&gt;&lt;/mfrac&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-128" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-129" class="mjx-mrow"><span id="MJXc-Node-130" class="mjx-msubsup"><span class="mjx-base" style="margin-right: -0.186em;"><span id="MJXc-Node-131" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.186em;">V</span></span></span><span class="mjx-sub" style="font-size: 70.7%; vertical-align: -0.219em; padding-right: 0.071em;"><span id="MJXc-Node-132" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.003em;">d</span></span></span></span><span id="MJXc-Node-133" class="mjx-mo MJXc-space3"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.064em; padding-bottom: 0.308em;">=</span></span><span id="MJXc-Node-134" class="mjx-mfrac MJXc-space3"><span class="mjx-box MJXc-stacked" style="width: 2.788em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 3.943em; top: -1.559em;"><span id="MJXc-Node-135" class="mjx-msubsup" style=""><span class="mjx-base" style="margin-right: -0.003em;"><span id="MJXc-Node-136" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.186em; padding-bottom: 0.308em; padding-right: 0.003em;">π</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0.076em; padding-right: 0.071em;"><span id="MJXc-Node-137" class="mjx-texatom" style=""><span id="MJXc-Node-138" class="mjx-mrow"><span id="MJXc-Node-139" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.003em;">d</span></span><span id="MJXc-Node-140" class="mjx-texatom"><span id="MJXc-Node-141" class="mjx-mrow"><span id="MJXc-Node-142" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.491em; padding-bottom: 0.614em;">/</span></span></span></span><span id="MJXc-Node-143" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">2</span></span></span></span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 3.943em; bottom: -1.891em;"><span id="MJXc-Node-144" class="mjx-mrow" style=""><span id="MJXc-Node-145" class="mjx-mi"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">Γ</span></span><span id="MJXc-Node-146" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.491em; padding-bottom: 0.614em;">(</span></span><span id="MJXc-Node-147" class="mjx-mstyle" style="font-size: 141.4%;"><span id="MJXc-Node-148" class="mjx-mrow"><span id="MJXc-Node-149" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 0.511em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 0.723em; top: -1.366em;"><span id="MJXc-Node-150" class="mjx-mi" style=""><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.003em;">d</span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 0.723em; bottom: -0.621em;"><span id="MJXc-Node-151" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">2</span></span></span><span style="border-bottom: 1.3px solid; top: -0.288em; width: 0.511em;" class="mjx-line"></span></span><span style="height: 1.405em; vertical-align: -0.439em;" class="mjx-vsize"></span></span></span></span><span id="MJXc-Node-152" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.308em; padding-bottom: 0.43em;">+</span></span><span id="MJXc-Node-153" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">1</span></span><span id="MJXc-Node-154" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.491em; padding-bottom: 0.614em;">)</span></span></span></span><span style="border-bottom: 1.3px solid; top: -0.288em; width: 2.788em;" class="mjx-line"></span></span><span style="height: 2.44em; vertical-align: -1.337em;" class="mjx-vsize"></span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><msub><mi>V</mi><mi>d</mi></msub><mo>=</mo><mfrac><msup><mi>π</mi><mrow class="MJX-TeXAtom-ORD"><mi>d</mi><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></mrow></msup><mrow><mi mathvariant="normal">Γ</mi><mo stretchy="false">(</mo><mstyle scriptlevel="0" displaystyle="false"><mfrac><mi>d</mi><mn>2</mn></mfrac></mstyle><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></mfrac></math></span></span><script type="math/tex" id="MathJax-Element-22">V_d = \frac{\pi^{d/2}}{\Gamma(\tfrac{d}{2}+1)}</script> in a plot. <em>Hint</em>: the Gamma function is available in <code>scipy.special.gamma</code>. <strong>(12 pts)</strong></p>
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<a name="comment-cell-82ea663199a69ee2"></a><a name="cell-82ea663199a69ee2"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right">
Score: 11.0 / 12.0 <a href="#top">(Top)</a>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="kn">import</span> <span class="nn">scipy.special</span>
<span class="k">def</span> <span class="nf">volume_d_ball</span><span class="p">(</span><span class="n">d</span><span class="p">):</span>
<span class="sd">"""Returns the volume of a d-dimensional unit ball."""</span>
<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">**</span><span class="p">(</span><span class="n">d</span><span class="o">/</span><span class="mi">2</span><span class="p">)</span><span class="o">/</span><span class="n">scipy</span><span class="o">.</span><span class="n">special</span><span class="o">.</span><span class="n">gamma</span><span class="p">(</span><span class="n">d</span><span class="o">/</span><span class="mi">2</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">10000</span>
<span class="n">d_range</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">(</span><span class="nb">range</span><span class="p">(</span><span class="mi">1</span><span class="p">,</span> <span class="mi">8</span><span class="p">))</span>
<span class="n">V_d_mc</span> <span class="o">=</span> <span class="p">[</span><span class="n">number_of_hits_in_d_dimensions</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">d</span><span class="p">)</span><span class="o">/</span><span class="n">N</span><span class="o">*</span><span class="mi">2</span><span class="o">**</span><span class="n">d</span> <span class="k">for</span> <span class="n">d</span> <span class="ow">in</span> <span class="n">d_range</span><span class="p">]</span>
<span class="c1">## Plotting</span>
<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">d_range</span><span class="p">,</span> <span class="n">V_d_mc</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"direct-sampling"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">d_range</span><span class="p">,</span> <span class="n">volume_d_ball</span><span class="p">(</span><span class="n">d_range</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">"exact"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"dimension $d$"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"volume $V_d$"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<b>Comments:</b> <p>Use dots</p>
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<h2 id="Efficiency-of-the-Metropolis-algorithm-and-the-1/2-rule">Efficiency of the Metropolis algorithm and the 1/2-rule<a class="anchor-link" href="#Efficiency-of-the-Metropolis-algorithm-and-the-1/2-rule"></a></h2><p>In
the lecture we mentioned the "1/2 rule" for Metropolis algorithms: if
moves are rarely rejected then typically you do not explore the domain
efficiently, while if most moves are rejected you are wasting efforts
proposing moves. A good rule of thumb then is to aim for a rejection
rate of <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-23-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;1&lt;/mn&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mo&gt;/&lt;/mo&gt;&lt;/mrow&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-155" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-156" class="mjx-mrow"><span id="MJXc-Node-157" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">1</span></span><span id="MJXc-Node-158" class="mjx-texatom"><span id="MJXc-Node-159" class="mjx-mrow"><span id="MJXc-Node-160" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.491em; padding-bottom: 0.614em;">/</span></span></span></span><span id="MJXc-Node-161" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">2</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>1</mn><mrow class="MJX-TeXAtom-ORD"><mo>/</mo></mrow><mn>2</mn></math></span></span><script type="math/tex" id="MathJax-Element-23">1/2</script>.
In this exercise you are asked to test whether this rule of thumb makes
any sense for the Markov-chain pebble game by varying the throwing
range <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-24-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03B4;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-162" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-163" class="mjx-mrow"><span id="MJXc-Node-164" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.007em;">δ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math></span></span><script type="math/tex" id="MathJax-Element-24">\delta</script>.</p>
<p><strong>(a)</strong> Estimate the mean square deviation <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-25-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mrow class=&quot;MJX-TeXAtom-ORD&quot;&gt;&lt;mi mathvariant=&quot;double-struck&quot;&gt;E&lt;/mi&gt;&lt;/mrow&gt;&lt;mo stretchy=&quot;false&quot;&gt;[&lt;/mo&gt;&lt;mo stretchy=&quot;false&quot;&gt;(&lt;/mo&gt;&lt;mstyle displaystyle=&quot;false&quot; scriptlevel=&quot;0&quot;&gt;&lt;mfrac&gt;&lt;mrow&gt;&lt;mn&gt;4&lt;/mn&gt;&lt;mtext&gt;hits&lt;/mtext&gt;&lt;/mrow&gt;&lt;mtext&gt;trials&lt;/mtext&gt;&lt;/mfrac&gt;&lt;/mstyle&gt;&lt;mo&gt;&amp;#x2212;&lt;/mo&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;)&lt;/mo&gt;&lt;mn&gt;2&lt;/mn&gt;&lt;/msup&gt;&lt;mo stretchy=&quot;false&quot;&gt;]&lt;/mo&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-165" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-166" class="mjx-mrow"><span id="MJXc-Node-167" class="mjx-texatom"><span id="MJXc-Node-168" class="mjx-mrow"><span id="MJXc-Node-169" class="mjx-mi"><span class="mjx-char MJXc-TeX-ams-R" style="padding-top: 0.43em; padding-bottom: 0.308em;">E</span></span></span></span><span id="MJXc-Node-170" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.491em; padding-bottom: 0.614em;">[</span></span><span id="MJXc-Node-171" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.491em; padding-bottom: 0.614em;">(</span></span><span id="MJXc-Node-172" class="mjx-mstyle"><span id="MJXc-Node-173" class="mjx-mrow"><span id="MJXc-Node-174" class="mjx-mfrac"><span class="mjx-box MJXc-stacked" style="width: 1.719em; padding: 0px 0.12em;"><span class="mjx-numerator" style="font-size: 70.7%; width: 2.431em; top: -1.367em;"><span id="MJXc-Node-175" class="mjx-mrow" style=""><span id="MJXc-Node-176" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">4</span></span><span id="MJXc-Node-177" class="mjx-mtext"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.43em; padding-bottom: 0.369em;">hits</span></span></span></span><span class="mjx-denominator" style="font-size: 70.7%; width: 2.431em; bottom: -0.66em;"><span id="MJXc-Node-178" class="mjx-mtext" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.43em; padding-bottom: 0.369em;">trials</span></span></span><span style="border-bottom: 1.3px solid; top: -0.288em; width: 1.719em;" class="mjx-line"></span></span><span style="height: 1.433em; vertical-align: -0.466em;" class="mjx-vsize"></span></span></span></span><span id="MJXc-Node-179" class="mjx-mo MJXc-space2"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.308em; padding-bottom: 0.43em;">−</span></span><span id="MJXc-Node-180" class="mjx-mi MJXc-space2"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.186em; padding-bottom: 0.308em; padding-right: 0.003em;">π</span></span><span id="MJXc-Node-181" class="mjx-msubsup"><span class="mjx-base"><span id="MJXc-Node-182" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.491em; padding-bottom: 0.614em;">)</span></span></span><span class="mjx-sup" style="font-size: 70.7%; vertical-align: 0.513em; padding-left: 0px; padding-right: 0.071em;"><span id="MJXc-Node-183" class="mjx-mn" style=""><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">2</span></span></span></span><span id="MJXc-Node-184" class="mjx-mo"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.491em; padding-bottom: 0.614em;">]</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mrow class="MJX-TeXAtom-ORD"><mi mathvariant="double-struck">E</mi></mrow><mo stretchy="false">[</mo><mo stretchy="false">(</mo><mstyle scriptlevel="0" displaystyle="false"><mfrac><mrow><mn>4</mn><mtext>hits</mtext></mrow><mtext>trials</mtext></mfrac></mstyle><mo>−</mo><mi>π</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup><mo stretchy="false">]</mo></math></span></span><script type="math/tex" id="MathJax-Element-25">\mathbb{E}[(\tfrac{4\text{hits}}{\text{trials}} - \pi)^2 ]</script> from <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-26-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03C0;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-185" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-186" class="mjx-mrow"><span id="MJXc-Node-187" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.186em; padding-bottom: 0.308em; padding-right: 0.003em;">π</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>π</mi></math></span></span><script type="math/tex" id="MathJax-Element-26">\pi</script> for different values of <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-27-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03B4;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-188" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-189" class="mjx-mrow"><span id="MJXc-Node-190" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.007em;">δ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math></span></span><script type="math/tex" id="MathJax-Element-27">\delta</script> ranging between <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-28-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;0&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-191" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-192" class="mjx-mrow"><span id="MJXc-Node-193" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">0</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>0</mn></math></span></span><script type="math/tex" id="MathJax-Element-28">0</script> and <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-29-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;3&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-194" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-195" class="mjx-mrow"><span id="MJXc-Node-196" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">3</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>3</mn></math></span></span><script type="math/tex" id="MathJax-Element-29">3</script>, but fixed number of trials (say, 2000). For a decent estimate of the mean you will need at least <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-30-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mn&gt;100&lt;/mn&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-197" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-198" class="mjx-mrow"><span id="MJXc-Node-199" class="mjx-mn"><span class="mjx-char MJXc-TeX-main-R" style="padding-top: 0.369em; padding-bottom: 0.369em;">100</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>100</mn></math></span></span><script type="math/tex" id="MathJax-Element-30">100</script> repetitions. <strong>(14 pts)</strong></p>
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<div class="prompt input_prompt">In&nbsp;[11]:</div><div class="panel panel-primary nbgrader_cell">
<a name="comment-cell-e3762269c892c3df"></a><a name="cell-e3762269c892c3df"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right">
Score: 14.0 / 14.0 <a href="#top">(Top)</a>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">N</span> <span class="o">=</span> <span class="mi">2000</span> <span class="c1"># number of trials</span>
<span class="n">m</span> <span class="o">=</span> <span class="mi">200</span> <span class="c1"># number of repetitions</span>
<span class="n">delta_range</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="c1"># throwing ranges</span>
<span class="n">p0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">])</span> <span class="c1"># starting position in origin</span>
<span class="n">hits</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">delta_range</span><span class="p">),</span> <span class="n">m</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)</span>
<span class="k">for</span> <span class="n">index_delta</span><span class="p">,</span> <span class="n">delta</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">delta_range</span><span class="p">):</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
<span class="n">hits</span><span class="p">[</span><span class="n">index_delta</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">=</span> <span class="n">markov_pebble</span><span class="p">(</span><span class="n">p0</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="n">N</span><span class="p">)</span>
<span class="c1">## Print the results</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="si">{:4}</span><span class="s2"> </span><span class="si">{:6}</span><span class="s2"> </span><span class="si">{:8}</span><span class="s2"> </span><span class="si">{:&lt;10}</span><span class="s2">"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="s1">'N'</span><span class="p">,</span><span class="s1">'delta'</span><span class="p">,</span><span class="s1">'4*hits/trials'</span><span class="p">,</span><span class="s1">'E[(4*hits/trials - pi)^2]'</span><span class="p">))</span>
<span class="k">for</span> <span class="n">index_delta</span><span class="p">,</span> <span class="n">delta</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">delta_range</span><span class="p">):</span>
<span class="n">pi_estimates</span> <span class="o">=</span> <span class="n">hits</span><span class="p">[</span><span class="n">index_delta</span><span class="p">]</span><span class="o">/</span><span class="n">N</span><span class="o">*</span><span class="mi">4</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="si">{:4}</span><span class="s2"> </span><span class="si">{:1.4f}</span><span class="s2"> </span><span class="si">{:1.4f}</span><span class="s2"> </span><span class="si">{:1.3f}</span><span class="s2">"</span>
<span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pi_estimates</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">pi_estimates</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">)))</span>
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<b>Comments:</b> <p>Nice formatting</p>
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<pre><span class="ansi-red-fg">---------------------------------------------------------------------------</span>
<span class="ansi-red-fg">KeyboardInterrupt</span> Traceback (most recent call last)
Input <span class="ansi-green-fg">In [11]</span>, in <span class="ansi-cyan-fg">&lt;cell line: 8&gt;</span><span class="ansi-blue-fg">()</span>
<span class="ansi-green-intense-fg ansi-bold"> 8</span> <span class="ansi-bold" style="color: rgb(0,135,0)">for</span> index_delta, delta <span class="ansi-bold" style="color: rgb(175,0,255)">in</span> <span style="color: rgb(0,135,0)">enumerate</span>(delta_range):
<span class="ansi-green-intense-fg ansi-bold"> 9</span> <span class="ansi-bold" style="color: rgb(0,135,0)">for</span> i <span class="ansi-bold" style="color: rgb(175,0,255)">in</span> <span style="color: rgb(0,135,0)">range</span>(m):
<span class="ansi-green-fg">---&gt; 10</span> hits[index_delta][i] <span style="color: rgb(98,98,98)">=</span> <span class="ansi-yellow-bg">markov_pebble</span><span class="ansi-yellow-bg">(</span><span class="ansi-yellow-bg">p0</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg">delta</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg">N</span><span class="ansi-yellow-bg">)</span>
<span class="ansi-green-intense-fg ansi-bold"> 12</span> <span style="color: rgb(95,135,135)">## Print the results</span>
<span class="ansi-green-intense-fg ansi-bold"> 13</span> <span style="color: rgb(0,135,0)">print</span>(<span style="color: rgb(175,0,0)">"</span><span class="ansi-bold" style="color: rgb(175,95,135)">{:4}</span><span style="color: rgb(175,0,0)"> </span><span class="ansi-bold" style="color: rgb(175,95,135)">{:6}</span><span style="color: rgb(175,0,0)"> </span><span class="ansi-bold" style="color: rgb(175,95,135)">{:8}</span><span style="color: rgb(175,0,0)"> </span><span class="ansi-bold" style="color: rgb(175,95,135)">{:&lt;10}</span><span style="color: rgb(175,0,0)">"</span><span style="color: rgb(98,98,98)">.</span>format(<span style="color: rgb(175,0,0)">'</span><span style="color: rgb(175,0,0)">N</span><span style="color: rgb(175,0,0)">'</span>,<span style="color: rgb(175,0,0)">'</span><span style="color: rgb(175,0,0)">delta</span><span style="color: rgb(175,0,0)">'</span>,<span style="color: rgb(175,0,0)">'</span><span style="color: rgb(175,0,0)">4*hits/trials</span><span style="color: rgb(175,0,0)">'</span>,<span style="color: rgb(175,0,0)">'</span><span style="color: rgb(175,0,0)">E[(4*hits/trials - pi)^2]</span><span style="color: rgb(175,0,0)">'</span>))
Input <span class="ansi-green-fg">In [2]</span>, in <span class="ansi-cyan-fg">markov_pebble</span><span class="ansi-blue-fg">(start, delta, N)</span>
<span class="ansi-green-intense-fg ansi-bold"> 70</span> position <span style="color: rgb(98,98,98)">=</span> start
<span class="ansi-green-intense-fg ansi-bold"> 71</span> <span class="ansi-bold" style="color: rgb(0,135,0)">for</span> i <span class="ansi-bold" style="color: rgb(175,0,255)">in</span> <span style="color: rgb(0,135,0)">range</span>(N):
<span class="ansi-green-fg">---&gt; 72</span> position <span style="color: rgb(98,98,98)">=</span> <span class="ansi-yellow-bg">sample_next_position</span><span class="ansi-yellow-bg">(</span><span class="ansi-yellow-bg">position</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg">delta</span><span class="ansi-yellow-bg">)</span>
<span class="ansi-green-intense-fg ansi-bold"> 73</span> <span class="ansi-bold" style="color: rgb(0,135,0)">if</span> is_in_circle(position):
<span class="ansi-green-intense-fg ansi-bold"> 74</span> number_hits <span style="color: rgb(98,98,98)">+</span><span style="color: rgb(98,98,98)">=</span> <span style="color: rgb(98,98,98)">1</span>
Input <span class="ansi-green-fg">In [2]</span>, in <span class="ansi-cyan-fg">sample_next_position</span><span class="ansi-blue-fg">(position, delta)</span>
<span class="ansi-green-intense-fg ansi-bold"> 59</span> <span class="ansi-bold" style="color: rgb(0,135,0)">def</span> <span style="color: rgb(0,0,255)">sample_next_position</span>(position,delta):
<span class="ansi-green-intense-fg ansi-bold"> 60</span> <span style="color: rgb(175,0,0)">"""Attempt a throw and reject when outside the square."""</span>
<span class="ansi-green-fg">---&gt; 61</span> next_position <span style="color: rgb(98,98,98)">=</span> position <span style="color: rgb(98,98,98)">+</span> delta<span style="color: rgb(98,98,98)">*</span><span class="ansi-yellow-bg">random_in_disk</span><span class="ansi-yellow-bg">(</span><span class="ansi-yellow-bg">)</span>
<span class="ansi-green-intense-fg ansi-bold"> 62</span> <span class="ansi-bold" style="color: rgb(0,135,0)">if</span> is_in_square(next_position):
<span class="ansi-green-intense-fg ansi-bold"> 63</span> <span class="ansi-bold" style="color: rgb(0,135,0)">return</span> next_position <span style="color: rgb(95,135,135)"># accept!</span>
Input <span class="ansi-green-fg">In [2]</span>, in <span class="ansi-cyan-fg">random_in_disk</span><span class="ansi-blue-fg">()</span>
<span class="ansi-green-intense-fg ansi-bold"> 23</span> <span class="ansi-bold" style="color: rgb(0,135,0)">def</span> <span style="color: rgb(0,0,255)">random_in_disk</span>():
<span class="ansi-green-intense-fg ansi-bold"> 24</span> <span style="color: rgb(175,0,0)">"""Returns a uniform point in the unit disk via rejection."""</span>
<span class="ansi-green-fg">---&gt; 25</span> position <span style="color: rgb(98,98,98)">=</span> <span class="ansi-yellow-bg">random_in_square</span><span class="ansi-yellow-bg">(</span><span class="ansi-yellow-bg">)</span>
<span class="ansi-green-intense-fg ansi-bold"> 26</span> <span class="ansi-bold" style="color: rgb(0,135,0)">while</span> <span class="ansi-bold" style="color: rgb(175,0,255)">not</span> is_in_circle(position):
<span class="ansi-green-intense-fg ansi-bold"> 27</span> position <span style="color: rgb(98,98,98)">=</span> random_in_square()
Input <span class="ansi-green-fg">In [2]</span>, in <span class="ansi-cyan-fg">random_in_square</span><span class="ansi-blue-fg">()</span>
<span class="ansi-green-intense-fg ansi-bold"> 7</span> <span class="ansi-bold" style="color: rgb(0,135,0)">def</span> <span style="color: rgb(0,0,255)">random_in_square</span>():
<span class="ansi-green-intense-fg ansi-bold"> 8</span> <span style="color: rgb(175,0,0)">"""Returns a random position in the square [-1,1)x[-1,1)."""</span>
<span class="ansi-green-fg">----&gt; 9</span> <span class="ansi-bold" style="color: rgb(0,135,0)">return</span> <span class="ansi-yellow-bg">rng</span><span class="ansi-yellow-bg" style="color: rgb(98,98,98)">.</span><span class="ansi-yellow-bg">uniform</span><span class="ansi-yellow-bg">(</span><span class="ansi-yellow-bg" style="color: rgb(98,98,98)">-</span><span class="ansi-yellow-bg" style="color: rgb(98,98,98)">1</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg" style="color: rgb(98,98,98)">1</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg" style="color: rgb(98,98,98)">2</span><span class="ansi-yellow-bg">)</span>
<span class="ansi-red-fg">KeyboardInterrupt</span>: </pre>
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<p><strong>(b)</strong> Measure the rejection rate for the same simulations as in (a). <strong>(14 pts)</strong></p>
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<div class="prompt input_prompt">In&nbsp;[12]:</div><div class="panel panel-primary nbgrader_cell">
<a name="comment-cell-9726c572b9859251"></a><a name="cell-9726c572b9859251"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right">
Score: 14.0 / 14.0 <a href="#top">(Top)</a>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># To calculate the rejection rate, we need to know the positions. I will copy</span>
<span class="c1"># and adapt the above code and the code in the markov_pebble function as is</span>
<span class="c1"># provided at the beginning of this document.</span>
<span class="n">N</span> <span class="o">=</span> <span class="mi">2000</span> <span class="c1"># number of trials</span>
<span class="n">m</span> <span class="o">=</span> <span class="mi">200</span> <span class="c1"># number of repetitions</span>
<span class="n">delta_range</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">linspace</span><span class="p">(</span><span class="mi">0</span><span class="p">,</span> <span class="mi">3</span><span class="p">,</span> <span class="mi">10</span><span class="p">)</span> <span class="c1"># throwing ranges</span>
<span class="n">p0</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">array</span><span class="p">([</span><span class="mf">0.</span><span class="p">,</span> <span class="mf">0.</span><span class="p">])</span> <span class="c1"># starting position in origin</span>
<span class="n">hits</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">delta_range</span><span class="p">),</span> <span class="n">m</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)</span>
<span class="n">rejections</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">zeros</span><span class="p">((</span><span class="nb">len</span><span class="p">(</span><span class="n">delta_range</span><span class="p">),</span> <span class="n">m</span><span class="p">),</span> <span class="n">dtype</span><span class="o">=</span><span class="nb">int</span><span class="p">)</span>
<span class="k">for</span> <span class="n">index_delta</span><span class="p">,</span> <span class="n">delta</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">delta_range</span><span class="p">):</span>
<span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">m</span><span class="p">):</span>
<span class="n">prev_position</span> <span class="o">=</span> <span class="kc">None</span>
<span class="k">for</span> <span class="n">position</span> <span class="ow">in</span> <span class="n">markov_pebble_generator</span><span class="p">(</span><span class="n">p0</span><span class="p">,</span><span class="n">delta</span><span class="p">,</span><span class="n">N</span><span class="p">):</span>
<span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">all</span><span class="p">(</span><span class="n">position</span> <span class="o">==</span> <span class="n">prev_position</span><span class="p">):</span>
<span class="n">rejections</span><span class="p">[</span><span class="n">index_delta</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="k">if</span> <span class="n">is_in_circle</span><span class="p">(</span><span class="n">position</span><span class="p">):</span>
<span class="n">hits</span><span class="p">[</span><span class="n">index_delta</span><span class="p">][</span><span class="n">i</span><span class="p">]</span> <span class="o">+=</span> <span class="mi">1</span>
<span class="n">prev_position</span> <span class="o">=</span> <span class="n">position</span>
<span class="c1">## Print the results</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="si">{:4}</span><span class="s2"> </span><span class="si">{:6}</span><span class="s2"> </span><span class="si">{:8}</span><span class="s2"> </span><span class="si">{:&lt;10}</span><span class="s2"> </span><span class="si">{:&lt;10}</span><span class="s2">"</span>
<span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="s1">'N'</span><span class="p">,</span><span class="s1">'delta'</span><span class="p">,</span><span class="s1">'4*hits/trials'</span><span class="p">,</span><span class="s1">'E[(4*hits/trials - pi)^2]'</span><span class="p">,</span> <span class="s1">'rejection rate'</span><span class="p">))</span>
<span class="k">for</span> <span class="n">index_delta</span><span class="p">,</span> <span class="n">delta</span> <span class="ow">in</span> <span class="nb">enumerate</span><span class="p">(</span><span class="n">delta_range</span><span class="p">):</span>
<span class="n">pi_estimates</span> <span class="o">=</span> <span class="n">hits</span><span class="p">[</span><span class="n">index_delta</span><span class="p">]</span><span class="o">/</span><span class="n">N</span><span class="o">*</span><span class="mi">4</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"</span><span class="si">{:4}</span><span class="s2"> </span><span class="si">{:1.4f}</span><span class="s2"> </span><span class="si">{:1.4f}</span><span class="s2"> </span><span class="si">{:1.3f}</span><span class="s2"> </span><span class="si">{:1.3f}</span><span class="s2">"</span>
<span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">delta</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">pi_estimates</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">pi_estimates</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">rejections</span><span class="p">[</span><span class="n">index_delta</span><span class="p">]</span><span class="o">/</span><span class="n">N</span><span class="p">)))</span>
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<b>Comments:</b> <p>Nice formatting</p>
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<pre><span class="ansi-red-fg">---------------------------------------------------------------------------</span>
<span class="ansi-red-fg">KeyboardInterrupt</span> Traceback (most recent call last)
Input <span class="ansi-green-fg">In [12]</span>, in <span class="ansi-cyan-fg">&lt;cell line: 13&gt;</span><span class="ansi-blue-fg">()</span>
<span class="ansi-green-intense-fg ansi-bold"> 15</span> prev_position <span style="color: rgb(98,98,98)">=</span> <span class="ansi-bold" style="color: rgb(0,135,0)">None</span>
<span class="ansi-green-intense-fg ansi-bold"> 16</span> <span class="ansi-bold" style="color: rgb(0,135,0)">for</span> position <span class="ansi-bold" style="color: rgb(175,0,255)">in</span> markov_pebble_generator(p0,delta,N):
<span class="ansi-green-fg">---&gt; 17</span> <span class="ansi-bold" style="color: rgb(0,135,0)">if</span> <span class="ansi-yellow-bg">np</span><span class="ansi-yellow-bg" style="color: rgb(98,98,98)">.</span><span class="ansi-yellow-bg">all</span><span class="ansi-yellow-bg">(</span><span class="ansi-yellow-bg">position</span><span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg" style="color: rgb(98,98,98)">==</span><span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg">prev_position</span><span class="ansi-yellow-bg">)</span>:
<span class="ansi-green-intense-fg ansi-bold"> 18</span> rejections[index_delta][i] <span style="color: rgb(98,98,98)">+</span><span style="color: rgb(98,98,98)">=</span> <span style="color: rgb(98,98,98)">1</span>
<span class="ansi-green-intense-fg ansi-bold"> 19</span> <span class="ansi-bold" style="color: rgb(0,135,0)">if</span> is_in_circle(position):
File <span class="ansi-green-fg">&lt;__array_function__ internals&gt;:5</span>, in <span class="ansi-cyan-fg">all</span><span class="ansi-blue-fg">(*args, **kwargs)</span>
File <span class="ansi-green-fg">/opt/jupyter-conda/lib/python3.9/site-packages/numpy/core/fromnumeric.py:2450</span>, in <span class="ansi-cyan-fg">all</span><span class="ansi-blue-fg">(a, axis, out, keepdims, where)</span>
<span class="ansi-green-intense-fg ansi-bold"> 2367</span> <span style="color: rgb(175,0,255)">@array_function_dispatch</span>(_all_dispatcher)
<span class="ansi-green-intense-fg ansi-bold"> 2368</span> <span class="ansi-bold" style="color: rgb(0,135,0)">def</span> <span style="color: rgb(0,0,255)">all</span>(a, axis<span style="color: rgb(98,98,98)">=</span><span class="ansi-bold" style="color: rgb(0,135,0)">None</span>, out<span style="color: rgb(98,98,98)">=</span><span class="ansi-bold" style="color: rgb(0,135,0)">None</span>, keepdims<span style="color: rgb(98,98,98)">=</span>np<span style="color: rgb(98,98,98)">.</span>_NoValue, <span style="color: rgb(98,98,98)">*</span>, where<span style="color: rgb(98,98,98)">=</span>np<span style="color: rgb(98,98,98)">.</span>_NoValue):
<span class="ansi-green-intense-fg ansi-bold"> 2369</span> <span style="color: rgb(175,0,0)">"""</span>
<span class="ansi-green-intense-fg ansi-bold"> 2370</span> <span style="color: rgb(175,0,0)"> Test whether all array elements along a given axis evaluate to True.</span>
<span class="ansi-green-intense-fg ansi-bold"> 2371</span>
<span class="ansi-green-fg"> (...)</span>
<span class="ansi-green-intense-fg ansi-bold"> 2448</span>
<span class="ansi-green-intense-fg ansi-bold"> 2449</span> <span style="color: rgb(175,0,0)"> """</span>
<span class="ansi-green-fg">-&gt; 2450</span> <span class="ansi-bold" style="color: rgb(0,135,0)">return</span> <span class="ansi-yellow-bg">_wrapreduction</span><span class="ansi-yellow-bg">(</span><span class="ansi-yellow-bg">a</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg">np</span><span class="ansi-yellow-bg" style="color: rgb(98,98,98)">.</span><span class="ansi-yellow-bg">logical_and</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg" style="color: rgb(175,0,0)">'</span><span class="ansi-yellow-bg" style="color: rgb(175,0,0)">all</span><span class="ansi-yellow-bg" style="color: rgb(175,0,0)">'</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg">axis</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg ansi-bold" style="color: rgb(0,135,0)">None</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg">out</span><span class="ansi-yellow-bg">,</span>
<span class="ansi-green-intense-fg ansi-bold"> 2451</span> <span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg">keepdims</span><span class="ansi-yellow-bg" style="color: rgb(98,98,98)">=</span><span class="ansi-yellow-bg">keepdims</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg">where</span><span class="ansi-yellow-bg" style="color: rgb(98,98,98)">=</span><span class="ansi-yellow-bg">where</span><span class="ansi-yellow-bg">)</span>
File <span class="ansi-green-fg">/opt/jupyter-conda/lib/python3.9/site-packages/numpy/core/fromnumeric.py:86</span>, in <span class="ansi-cyan-fg">_wrapreduction</span><span class="ansi-blue-fg">(obj, ufunc, method, axis, dtype, out, **kwargs)</span>
<span class="ansi-green-intense-fg ansi-bold"> 83</span> <span class="ansi-bold" style="color: rgb(0,135,0)">else</span>:
<span class="ansi-green-intense-fg ansi-bold"> 84</span> <span class="ansi-bold" style="color: rgb(0,135,0)">return</span> reduction(axis<span style="color: rgb(98,98,98)">=</span>axis, out<span style="color: rgb(98,98,98)">=</span>out, <span style="color: rgb(98,98,98)">*</span><span style="color: rgb(98,98,98)">*</span>passkwargs)
<span class="ansi-green-fg">---&gt; 86</span> <span class="ansi-bold" style="color: rgb(0,135,0)">return</span> <span class="ansi-yellow-bg">ufunc</span><span class="ansi-yellow-bg" style="color: rgb(98,98,98)">.</span><span class="ansi-yellow-bg">reduce</span><span class="ansi-yellow-bg">(</span><span class="ansi-yellow-bg">obj</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg">axis</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg">dtype</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg">out</span><span class="ansi-yellow-bg">,</span><span class="ansi-yellow-bg"> </span><span class="ansi-yellow-bg" style="color: rgb(98,98,98)">*</span><span class="ansi-yellow-bg" style="color: rgb(98,98,98)">*</span><span class="ansi-yellow-bg">passkwargs</span><span class="ansi-yellow-bg">)</span>
<span class="ansi-red-fg">KeyboardInterrupt</span>: </pre>
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<p><strong>(c)</strong> Plot both the mean square deviation and the rejection rate as function of <span class="MathJax_Preview" style="color: inherit;"></span><span id="MathJax-Element-31-Frame" class="mjx-chtml MathJax_CHTML" tabindex="0" style="font-size: 117%; position: relative;" data-mathml="&lt;math xmlns=&quot;http://www.w3.org/1998/Math/MathML&quot;&gt;&lt;mi&gt;&amp;#x03B4;&lt;/mi&gt;&lt;/math&gt;" role="presentation"><span id="MJXc-Node-200" class="mjx-math" aria-hidden="true"><span id="MJXc-Node-201" class="mjx-mrow"><span id="MJXc-Node-202" class="mjx-mi"><span class="mjx-char MJXc-TeX-math-I" style="padding-top: 0.491em; padding-bottom: 0.308em; padding-right: 0.007em;">δ</span></span></span></span><span class="MJX_Assistive_MathML" role="presentation"><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>δ</mi></math></span></span><script type="math/tex" id="MathJax-Element-31">\delta</script>. How well does the 1/2 rule apply in this situation? <strong>(12 pts)</strong></p>
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<div class="prompt input_prompt">In&nbsp;[13]:</div><div class="panel panel-primary nbgrader_cell">
<a name="comment-cell-0677a8cb2d72d30f"></a><a name="cell-0677a8cb2d72d30f"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right">
Score: 10.0 / 12.0 <a href="#top">(Top)</a>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">title</span><span class="p">(</span><span class="s2">"Mean square deviation and rejection rate of $</span><span class="se">\\</span><span class="s2">pi$ guesses by the Metropolis algorithm for </span><span class="si">{}</span><span class="s2"> trials of each </span><span class="si">{}</span><span class="s2"> repetitions"</span><span class="o">.</span><span class="n">format</span><span class="p">(</span><span class="n">N</span><span class="p">,</span> <span class="n">m</span><span class="p">))</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">delta_range</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">((</span><span class="n">hits</span><span class="o">/</span><span class="n">N</span><span class="o">*</span><span class="mi">4</span> <span class="o">-</span> <span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">"$\mathbb</span><span class="si">{E}</span><span class="s2">[(</span><span class="se">\\</span><span class="s2">frac</span><span class="si">{4hits}{trials}</span><span class="s2"> - </span><span class="se">\\</span><span class="s2">pi)^2]$"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">delta_range</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">rejections</span><span class="o">/</span><span class="n">N</span><span class="p">,</span> <span class="n">axis</span><span class="o">=</span><span class="mi">1</span><span class="p">),</span> <span class="n">label</span><span class="o">=</span><span class="s2">"rejection rate"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"throwing range $</span><span class="se">\\</span><span class="s2">delta$"</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
<span class="n">plt</span><span class="o">.</span><span class="n">rc</span><span class="p">(</span><span class="s1">'legend'</span><span class="p">,</span> <span class="n">fontsize</span><span class="o">=</span><span class="mi">13</span><span class="p">)</span>
<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"The rejection rate is around 1/2 for delta = 1.3333, which also yields near lowest mean square deviation to pi."</span><span class="p">)</span>
<span class="nb">print</span><span class="p">(</span><span class="s2">"The rule of thumb works quite nicely in this case, although it does not yields the best result."</span><span class="p">)</span>
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<b>Comments:</b> <p>Good labels. Due to the outlier at 0, we cannot see the minimum MSD in the plot.</p>
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<pre>The rejection rate is around 1/2 for delta = 1.3333, which also yields near lowest mean square deviation to pi.
The rule of thumb works quite nicely in this case, although it does not yields the best result.
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