1028 lines
202 KiB
HTML
1028 lines
202 KiB
HTML
<!DOCTYPE html>
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<html>
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<head>
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<meta charset="utf-8" />
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<title>exercise_sheet_03</title>
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<style type="text/css">
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/*!
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* Bootstrap v3.3.4 (http://getbootstrap.com)
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* Copyright 2011-2015 Twitter, Inc.
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* Licensed under MIT (https://github.com/twbs/bootstrap/blob/master/LICENSE)
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<h4>exercise_sheet_03 (Score: 49.0 / 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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<ul>
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<li>For your convenience functions from the lecture are included below. Feel free to reuse them without copying to the exercise solution box.</li>
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<li>For each part of the exercise a solution box has been added, but you may insert additional boxes. Do not hesitate to add Markdown boxes for textual or LaTeX answers (via <code>Cell > Cell Type > Markdown</code>). But make sure to replace any part that says <code>YOUR CODE HERE</code> or <code>YOUR ANSWER HERE</code> and remove the <code>raise NotImplementedError()</code>.</li>
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<li>Please make your code readable by humans (and not just by the Python interpreter): choose informative function and variable names and use consistent formatting. Feel free to check the <a href="https://www.python.org/dev/peps/pep-0008/">PEP 8 Style Guide for Python</a> for the widely adopted coding conventions or <a href="https://realpython.com/python-pep8/">this guide for explanation</a>.</li>
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<li>Make sure that the full notebook runs without errors before submitting your work. This you can do by selecting <code>Kernel > Restart & Run All</code> in the jupyter menu.</li>
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<li>For some exercises test cases have been provided in a separate cell in the form of <code>assert</code> statements. When run, a successful test will give no output, whereas a failed test will display an error message.</li>
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<li>Each sheet has 100 points worth of exercises. Note that only the grades of sheets number 2, 4, 6, 8 count towards the course examination. Submitting sheets 1, 3, 5, 7 & 9 is voluntary and their grades are just for feedback.</li>
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</ul>
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<p>Please fill in your name here:</p>
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<div class="prompt input_prompt">In [1]:</div><div class="inner_cell">
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<div class="input_area">
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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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<span class="n">NAMES_OF_COLLABORATORS</span> <span class="o">=</span> <span class="s2">""</span>
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</pre></div>
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<p><strong>Exercise sheet 3</strong></p>
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<p>Code from the lectures:</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>
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<span class="kn">import</span> <span class="nn">matplotlib.pylab</span> <span class="k">as</span> <span class="nn">plt</span>
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<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>
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<span class="o">%</span><span class="k">matplotlib</span> inline
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<span class="k">def</span> <span class="nf">sample_acceptance_rejection</span><span class="p">(</span><span class="n">sample_z</span><span class="p">,</span><span class="n">accept_probability</span><span class="p">):</span>
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<span class="k">while</span> <span class="kc">True</span><span class="p">:</span>
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<span class="n">x</span> <span class="o">=</span> <span class="n">sample_z</span><span class="p">()</span>
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<span class="k">if</span> <span class="n">rng</span><span class="o">.</span><span class="n">random</span><span class="p">()</span> <span class="o"><</span> <span class="n">accept_probability</span><span class="p">(</span><span class="n">x</span><span class="p">):</span>
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<span class="k">return</span> <span class="n">x</span>
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<span class="k">def</span> <span class="nf">estimate_expectation</span><span class="p">(</span><span class="n">sampler</span><span class="p">,</span><span class="n">n</span><span class="p">):</span>
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<span class="sd">'''Compute beste estimate of mean and 1-sigma error with n samples.'''</span>
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<span class="n">samples</span> <span class="o">=</span> <span class="p">[</span><span class="n">sampler</span><span class="p">()</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="n">n</span><span class="p">)]</span>
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<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">mean</span><span class="p">(</span><span class="n">samples</span><span class="p">),</span> <span class="n">np</span><span class="o">.</span><span class="n">std</span><span class="p">(</span><span class="n">samples</span><span class="p">)</span><span class="o">/</span><span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span>
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<span class="k">def</span> <span class="nf">estimate_expectation_one_pass</span><span class="p">(</span><span class="n">sampler</span><span class="p">,</span><span class="n">n</span><span class="p">):</span>
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<span class="n">sample_mean</span> <span class="o">=</span> <span class="n">sample_square_dev</span> <span class="o">=</span> <span class="mf">0.0</span>
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<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">1</span><span class="p">,</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">):</span>
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<span class="n">delta</span> <span class="o">=</span> <span class="n">sampler</span><span class="p">()</span> <span class="o">-</span> <span class="n">sample_mean</span>
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<span class="n">sample_mean</span> <span class="o">+=</span> <span class="n">delta</span> <span class="o">/</span> <span class="n">k</span>
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<span class="n">sample_square_dev</span> <span class="o">+=</span> <span class="p">(</span><span class="n">k</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span><span class="o">*</span><span class="n">delta</span><span class="o">*</span><span class="n">delta</span><span class="o">/</span><span class="n">k</span>
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<span class="k">return</span> <span class="n">sample_mean</span><span class="p">,</span> <span class="n">np</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">sample_square_dev</span> <span class="o">/</span> <span class="p">(</span><span class="n">n</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="mi">1</span><span class="p">)))</span>
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</pre></div>
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<h2 id="Acceptance-rejection-sampling">Acceptance-rejection sampling<a class="anchor-link" href="#Acceptance-rejection-sampling">¶</a></h2><p><strong>(35 points)</strong></p>
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<p>The goal of this exercise is to develop a fast sampling algorithm of the discrete random variable $X$ with probability mass function $$p_X(k) = \frac{6}{\pi^2} k^{-2}, \qquad k=1,2,\ldots$$</p>
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<p><strong>(a)</strong> Let $Z$ be the discrete random variable with $p_Z(k) = \frac{1}{k} - \frac{1}{k+1}$ for $k=1,2,\ldots$. Write a function to compute the inverse CDF $F_Z^{-1}(u)$, such that you can use the inversion method to sample $Z$ efficiently. <strong>(15 pts)</strong></p>
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<a name="comment-cell-b0b3fcf6d9bf35a0"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right"><a href="#top">(Top)</a></span></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">f_inverse_Z</span><span class="p">(</span><span class="n">u</span><span class="p">):</span>
|
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<span class="sd">'''Compute the inverse CDF of Z, i.e. F_Z^{-1}(u) for 0 <= u <= 1.'''</span>
|
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<span class="k">assert</span> <span class="mi">0</span> <span class="o"><=</span> <span class="n">u</span> <span class="ow">and</span> <span class="n">u</span> <span class="o"><=</span> <span class="mi">1</span>
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<span class="c1"># we need to round to return a natural number (excluding zero, yes).</span>
|
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<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">ceil</span><span class="p">(</span><span class="n">u</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">u</span><span class="p">))</span><span class="o">.</span><span class="n">astype</span><span class="p">(</span><span class="nb">int</span><span class="p">)</span>
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<span class="k">def</span> <span class="nf">random_Z</span><span class="p">():</span>
|
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<span class="k">return</span> <span class="nb">int</span><span class="p">(</span><span class="n">f_inverse_Z</span><span class="p">(</span><span class="n">rng</span><span class="o">.</span><span class="n">random</span><span class="p">()))</span> <span class="c1"># make sure to return an integer</span>
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</pre></div>
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<div class="prompt input_prompt">In [4]:</div><div class="panel panel-primary nbgrader_cell">
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<a name="cell-a28cb07f90510b94"></a><div class="panel-heading"><span class="nbgrader-label">Grade cell: <code>cell-a28cb07f90510b94</code></span>
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<span class="pull-right">
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Score: 15.0 / 15.0 <a href="#top">(Top)</a>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">assert</span> <span class="n">f_inverse_Z</span><span class="p">(</span><span class="mf">0.2</span><span class="p">)</span><span class="o">==</span><span class="mi">1</span>
|
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<span class="k">assert</span> <span class="n">f_inverse_Z</span><span class="p">(</span><span class="mf">0.51</span><span class="p">)</span><span class="o">==</span><span class="mi">2</span>
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<span class="k">assert</span> <span class="n">f_inverse_Z</span><span class="p">(</span><span class="mf">0.76</span><span class="p">)</span><span class="o">==</span><span class="mi">4</span>
|
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<span class="k">assert</span> <span class="n">f_inverse_Z</span><span class="p">(</span><span class="mf">0.991</span><span class="p">)</span><span class="o">==</span><span class="mi">111</span>
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</pre></div>
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<p><strong>(b)</strong> Implement a sampler for $X$ using acceptance-rejection based on the sampler of $Z$. For this you need to first determine a $c$ such that $p_X(k) \leq c\,p_Z(k)$ for all $k=1,2,\ldots$, and then consider an acceptance probability $p_X(k) / (c p_Z(k))$. Verify the validity of your sampler numerically (e.g. for $k=1,\ldots,10$). <strong>(20 pts)</strong></p>
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<div class="prompt input_prompt">In [5]:</div><div class="panel panel-primary nbgrader_cell">
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<a name="comment-cell-20fd55e7a4c4c531"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right"><a href="#top">(Top)</a></span></div>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">accept_probability_X</span><span class="p">(</span><span class="n">k</span><span class="p">):</span>
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<span class="sd">'''Return the appropriate acceptance probability on the event Z=k.'''</span>
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<span class="c1"># The resulting c is found by doing algebraic things, and seeing that</span>
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<span class="c1"># 6/pi^2*(1 + 1/k) <= c should hold for all allowed k. For k = 1, c is</span>
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<span class="c1"># the largest, so: tada.</span>
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<span class="n">c</span> <span class="o">=</span> <span class="mi">12</span><span class="o">*</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="o">-</span><span class="mi">2</span><span class="p">)</span>
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<span class="c1"># But we do not need this variable, we only need the acceptance probability.</span>
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<span class="k">return</span> <span class="mf">.5</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">+</span> <span class="mi">1</span><span class="o">/</span><span class="n">k</span><span class="p">)</span>
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<span class="k">def</span> <span class="nf">random_X</span><span class="p">():</span>
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<span class="k">return</span> <span class="n">sample_acceptance_rejection</span><span class="p">(</span><span class="n">random_Z</span><span class="p">,</span><span class="n">accept_probability_X</span><span class="p">)</span>
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<span class="c1"># Verify numerically</span>
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<span class="n">samples</span> <span class="o">=</span> <span class="p">[</span><span class="n">random_X</span><span class="p">()</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">1000000</span><span class="p">)]</span>
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<span class="n">k</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">11</span><span class="p">))</span>
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<span class="c1"># The theoretical densities to compare the densities from the</span>
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<span class="c1"># acceptance-rejection densities to:</span>
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<span class="n">p_X</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">k</span><span class="p">:</span> <span class="mi">6</span><span class="o">/</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">k</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">figure</span><span class="p">()</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">hist</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">k</span> <span class="o">-</span> <span class="mf">.5</span><span class="p">,</span> <span class="n">density</span><span class="o">=</span><span class="kc">True</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"lightblue"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"density of acceptance-rejection samples"</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">scatter</span><span class="p">(</span><span class="n">k</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">],</span> <span class="p">[</span><span class="n">p_X</span><span class="p">(</span><span class="n">i</span><span class="p">)</span> <span class="k">for</span> <span class="n">i</span> <span class="ow">in</span> <span class="n">k</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]],</span> <span class="n">s</span><span class="o">=</span><span class="mi">800</span><span class="p">,</span> <span class="n">marker</span><span class="o">=</span><span class="s2">"_"</span><span class="p">,</span> <span class="n">color</span><span class="o">=</span><span class="s2">"black"</span><span class="p">,</span> <span class="n">label</span><span class="o">=</span><span class="s2">"theoretical"</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">yscale</span><span class="p">(</span><span class="s2">"log"</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">ylabel</span><span class="p">(</span><span class="s2">"$p_X(k)$"</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">xlabel</span><span class="p">(</span><span class="s2">"$k$"</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">legend</span><span class="p">()</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">show</span><span class="p">()</span>
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<span class="c1"># TODO: The histogram overshoots the theoretical value every time.</span>
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</pre></div>
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</div>
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</div>
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<div class="panel-footer">
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<div>
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<b>Comments:</b> <p>Pas op met <code>density=True</code> in <code>plt.hist</code>. Het neemt bij het normaliseren niet de datapunten buiten de bins mee, en geeft daarom een iets onjuiste normalisering.</p>
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<div class="prompt input_prompt">In [6]:</div><div class="panel panel-primary nbgrader_cell">
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<a name="cell-04bad69f7cc2ad18"></a><div class="panel-heading"><span class="nbgrader-label">Grade cell: <code>cell-04bad69f7cc2ad18</code></span>
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Score: 19.0 / 20.0 <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>
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<span class="k">assert</span> <span class="nb">min</span><span class="p">([</span><span class="n">random_X</span><span class="p">()</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10000</span><span class="p">)])</span> <span class="o">>=</span> <span class="mi">1</span>
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<span class="n">assert_almost_equal</span><span class="p">([</span><span class="n">random_X</span><span class="p">()</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10000</span><span class="p">)]</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="mi">1</span><span class="p">),</span><span class="mi">6079</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mi">400</span><span class="p">)</span>
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<span class="n">assert_almost_equal</span><span class="p">([</span><span class="n">random_X</span><span class="p">()</span> <span class="k">for</span> <span class="n">_</span> <span class="ow">in</span> <span class="nb">range</span><span class="p">(</span><span class="mi">10000</span><span class="p">)]</span><span class="o">.</span><span class="n">count</span><span class="p">(</span><span class="mi">3</span><span class="p">),</span><span class="mi">675</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mi">75</span><span class="p">)</span>
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<h2 id="Monte-Carlo-integration-&-Importance-sampling">Monte Carlo integration & Importance sampling<a class="anchor-link" href="#Monte-Carlo-integration-&-Importance-sampling">¶</a></h2><p><strong>(30 Points)</strong></p>
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<p>Consider the integral</p>
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$$
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I = \int_0^1 \sin(\pi x(1-x))\mathrm{d}x = \mathbb{E}[X], \quad X=g(U), \quad g(U)=\sin(\pi U(1-U)),
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$$<p></p>
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<p>where $U$ is a uniform random variable in $(0,1)$.</p>
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<p><strong>(a)</strong> Use Monte Carlo integration based on sampling $U$ to estimate $I$ with $1\sigma$ error at most $0.001$. How many samples do you need? (It is not necessary to automate this: trial and error is sufficient.) <strong>(10 pts)</strong></p>
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<div class="prompt input_prompt">In [7]:</div><div class="panel panel-primary nbgrader_cell">
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<a name="comment-cell-47eab0537d21690e"></a><a name="cell-47eab0537d21690e"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right">
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Score: 10.0 / 10.0 <a href="#top">(Top)</a>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">sample_X</span><span class="p">():</span>
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<span class="n">x</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>
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<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">x</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">x</span><span class="p">))</span>
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<span class="n">n</span> <span class="o">=</span> <span class="mi">43402</span>
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<span class="n">sample_mean</span><span class="p">,</span> <span class="n">sample_stdev</span> <span class="o">=</span> <span class="n">estimate_expectation</span><span class="p">(</span><span class="n">sample_X</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="s2">"""</span>
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<span class="s2">After trial and error, I found that </span><span class="si">{}</span><span class="s2"> trials were sufficient to arrive at an 1-sigma error of around 0.001,</span>
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<span class="s2">with mean </span><span class="si">{:.6f}</span><span class="s2"> and standard deviation sigma </span><span class="si">{:.6f}</span><span class="s2">. Wolfram Alpha tells me it should be approximately I =</span>
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<span class="si">{}</span><span class="s2">, so I am happy.</span>
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<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">sample_mean</span><span class="p">,</span> <span class="n">sample_stdev</span><span class="p">,</span> <span class="mf">0.487595</span><span class="p">))</span>
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<span class="c1"># TODO: Maybe I did not answer the question. Reading is difficult.</span>
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</pre></div>
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<pre>
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After trial and error, I found that 43402 trials were sufficient to arrive at an 1-sigma error of around 0.001,
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with mean 0.487735 and standard deviation sigma 0.000995. Wolfram Alpha tells me it should be approximately I =
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0.487595, so I am happy.
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<p><strong>(b)</strong> Choose a random variable $Z$ on $(0,1)$ whose density resembles the integrand of $I$ and which you know how to sample efficiently (by inversion method, acceptance-rejection, or a built-in Python function). Estimate $I$ again using importance sampling, i.e. $I = \mathbb{E}[X']$ where $X' = g(Z) f_U(Z)/f_Z(Z)$, with an error of at most 0.001. How many samples did you need this time? <strong>(20 pts)</strong></p>
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<p>After hours of thinking, I tried $$f_Z(x) = \sec^2(\pi{}x) = \frac{1}{\cos^2(\pi{}x)}$$ on $(0, 1)$.
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It has cumulative density $$F_Z(x) = \int_0^x\sec^2(\pi{}t)dt = \left[ \frac{\tan (\pi{}t)}{\pi} \right]_{t=0}^x = \frac{\tan (\pi{}x)}{\pi}.$$ This density gives us $$X' := g(Z)f_U(Z)/f_Z(Z) = \sin(\pi{}Z(1-Z))\sin^2(\pi{}Z)$$ for $Z$ on $(0, 1)$. $Z$ can be sampled using the inversion method as $$F_Z^{-1}(p) = \frac{\arctan(\pi{}p)}{\pi}.$$</p>
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<p>Unluckily, I found that the function is way too non-constant.</p>
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<div class="prompt input_prompt">In [8]:</div><div class="panel panel-primary nbgrader_cell">
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<a name="comment-cell-467d7240beab5b29"></a><a name="cell-467d7240beab5b29"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right">
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Score: 5.0 / 20.0 <a href="#top">(Top)</a>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">sample_nice_Z</span><span class="p">():</span>
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<span class="sd">'''Sample from the nice distribution Z'''</span>
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<span class="n">p</span> <span class="o">=</span> <span class="n">rng</span><span class="o">.</span><span class="n">random</span><span class="p">()</span>
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<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">arctan</span><span class="p">(</span><span class="n">p</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="n">np</span><span class="o">.</span><span class="n">pi</span>
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<span class="k">def</span> <span class="nf">sample_X_prime</span><span class="p">():</span>
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<span class="sd">'''Sample from X'.'''</span>
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<span class="n">z</span> <span class="o">=</span> <span class="n">sample_nice_Z</span><span class="p">()</span>
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<span class="k">return</span> <span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span> <span class="o">-</span> <span class="n">z</span><span class="p">))</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">cos</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="o">*</span><span class="n">z</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span>
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<span class="n">n</span> <span class="o">=</span> <span class="mi">43402</span>
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<span class="n">sample_mean</span><span class="p">,</span> <span class="n">sample_stdev</span> <span class="o">=</span> <span class="n">estimate_expectation</span><span class="p">(</span><span class="n">sample_X_prime</span><span class="p">,</span> <span class="n">n</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="s2">"""</span>
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<span class="s2">After trial and error, I found that </span><span class="si">{}</span><span class="s2"> trials were sufficient to arrive at an 1-sigma error of around 0.001,</span>
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<span class="s2">with mean </span><span class="si">{:.6f}</span><span class="s2"> and standard deviation sigma </span><span class="si">{:.6f}</span><span class="s2">. Wolfram Alpha tells me it should be approximately I =</span>
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<span class="si">{}</span><span class="s2">, so I am happy.</span>
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<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">sample_mean</span><span class="p">,</span> <span class="n">sample_stdev</span><span class="p">,</span> <span class="mf">0.487595</span><span class="p">))</span>
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</pre></div>
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<b>Comments:</b> <p>Let op: integraal is alleen geldig voor x<1/2. Daarna heb je een divergentie die je niet correct afhandelt. Daardoor klopt je sampler niet.
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Let ook op je normalisatie van f_Z</p>
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<pre>
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After trial and error, I found that 43402 trials were sufficient to arrive at an 1-sigma error of around 0.001,
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with mean 0.175168 and standard deviation sigma 0.000417. Wolfram Alpha tells me it should be approximately I =
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0.487595, so I am happy.
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<h2 id="Direct-sampling-of-Dyck-paths">Direct sampling of Dyck paths<a class="anchor-link" href="#Direct-sampling-of-Dyck-paths">¶</a></h2><p><strong>(35 points)</strong></p>
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<p>Direct sampling of random variables in high dimensions requires some luck and/or ingenuity. Here is an example of a probability distribution on $\mathbb{Z}^{2n+1}$ that features prominently in the combinatorial literature and can be sampled directly in an efficient manner. A sequence $\mathbf{x}\equiv(x_0,x_1,\ldots,x_{2n})\in\mathbb{Z}^{2n+1}$ is said to be a <strong>Dyck path</strong> if $x_0=x_{2n}=0$, $x_i \geq 0$ and $|x_{i}-x_{i-1}|=1$ for all $i=1,\ldots,2n$. Dyck paths are counted by the Catalan numbers $C(n) = \frac{1}{n+1}\binom{2n}{n}$. Let $\mathbf{X}=(X_0,\ldots,X_n)$ be a <strong>uniform Dyck path</strong>, i.e. a random variable with probability mass function $p_{\mathbf{X}}(\mathbf{x}) = 1/C(n)$ for every Dyck path $\mathbf{x}$. Here is one way to sample $\mathbf{X}$.</p>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="k">def</span> <span class="nf">random_dyck_path</span><span class="p">(</span><span class="n">n</span><span class="p">):</span>
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<span class="sd">'''Returns a uniform Dyck path of length 2n as an array [x_0, x_1, ..., x_{2n}] of length 2n.'''</span>
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<span class="c1"># produce a (2n+1)-step random walk from 0 to -1</span>
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<span class="n">increments</span> <span class="o">=</span> <span class="p">[</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="n">n</span> <span class="o">+</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">*</span><span class="p">(</span><span class="n">n</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
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<span class="n">rng</span><span class="o">.</span><span class="n">shuffle</span><span class="p">(</span><span class="n">increments</span><span class="p">)</span>
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<span class="n">unconstrained_walk</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">increments</span><span class="p">)</span>
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<span class="c1"># determine the first time it reaches its minimum</span>
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<span class="n">argmin</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">argmin</span><span class="p">(</span><span class="n">unconstrained_walk</span><span class="p">)</span>
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<span class="c1"># cyclically permute the increments to ensure walk stays non-negative until last step</span>
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<span class="n">rotated_increments</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">roll</span><span class="p">(</span><span class="n">increments</span><span class="p">,</span><span class="o">-</span><span class="n">argmin</span><span class="p">)</span>
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<span class="c1"># turn off the superfluous -1 step</span>
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<span class="n">rotated_increments</span><span class="p">[</span><span class="mi">0</span><span class="p">]</span> <span class="o">=</span> <span class="mi">0</span>
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<span class="c1"># produce dyck path from increments</span>
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<span class="n">walk</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">cumsum</span><span class="p">(</span><span class="n">rotated_increments</span><span class="p">)</span>
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<span class="k">return</span> <span class="n">walk</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">plot</span><span class="p">(</span><span class="n">random_dyck_path</span><span class="p">(</span><span class="mi">50</span><span class="p">))</span>
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<p><strong>(a)</strong> Let $H$ be the (maximal) height of $X$, i.e. $H=\max_i X_i$. Estimate the expected height $\mathbb{E}[H]$ for $n = 2^5, 2^6, \ldots, 2^{11}$ (including error bars). Determine the growth $\mathbb{E}[H] \approx a\,n^\beta$ via an appropriate fit. <em>Hint</em>: use the <code>scipy.optimize.curve_fit</code> function with the option <code>sigma = ...</code> to incorporate the standard errors on $\mathbb{E}[H]$ in the fit. Note that when you supply the errors appropriately, fitting on linear or logarithmic scale should result in the same answer. <strong>(25 pts)</strong></p>
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<div class="prompt input_prompt">In [10]:</div><div class="panel panel-primary nbgrader_cell">
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<a name="comment-cell-310664cb8eaaa831"></a><a name="cell-310664cb8eaaa831"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right">
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Score: 0.0 / 10.0 <a href="#top">(Top)</a>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Collect height estimates</span>
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<span class="n">n_values</span> <span class="o">=</span> <span class="p">[</span><span class="mi">2</span><span class="o">**</span><span class="n">k</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">5</span><span class="p">,</span><span class="mi">11</span><span class="o">+</span><span class="mi">1</span><span class="p">)]</span>
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<span class="c1"># YOUR CODE HERE</span>
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<span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
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</pre></div>
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<b>Comments:</b> <p>No response.</p>
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<pre>
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<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
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<span class="ansi-red-fg">NotImplementedError</span> Traceback (most recent call last)
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Input <span class="ansi-green-fg">In [10]</span>, in <span class="ansi-cyan-fg"><cell line: 4></span><span class="ansi-blue-fg">()</span>
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<span class="ansi-green-intense-fg ansi-bold"> 2</span> n_values <span style="color: rgb(98,98,98)">=</span> [<span style="color: rgb(98,98,98)">2</span><span style="color: rgb(98,98,98)">*</span><span style="color: rgb(98,98,98)">*</span>k <span class="ansi-bold" style="color: rgb(0,135,0)">for</span> k <span class="ansi-bold" style="color: rgb(175,0,255)">in</span> <span style="color: rgb(0,135,0)">range</span>(<span style="color: rgb(98,98,98)">5</span>,<span style="color: rgb(98,98,98)">11</span><span style="color: rgb(98,98,98)">+</span><span style="color: rgb(98,98,98)">1</span>)]
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<span class="ansi-green-intense-fg ansi-bold"> 3</span> <span style="color: rgb(95,135,135)"># YOUR CODE HERE</span>
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<span class="ansi-green-fg">----> 4</span> <span class="ansi-bold" style="color: rgb(0,135,0)">raise</span> <span class="ansi-bold" style="color: rgb(215,95,95)">NotImplementedError</span>()
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<span class="ansi-red-fg">NotImplementedError</span>: </pre>
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<div class="prompt input_prompt">In [11]:</div><div class="panel panel-primary nbgrader_cell">
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<a name="comment-cell-0ca47ced5547d67c"></a><a name="cell-0ca47ced5547d67c"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right">
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Score: 0.0 / 10.0 <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">scipy.optimize</span> <span class="kn">import</span> <span class="n">curve_fit</span>
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<span class="c1"># Fitting</span>
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<span class="c1"># YOUR CODE HERE</span>
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<span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
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<span class="nb">print</span><span class="p">(</span><span class="s2">"Fit parameters: a = </span><span class="si">{}</span><span class="s2">, beta = </span><span class="si">{}</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">beta_fit</span><span class="p">))</span>
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<b>Comments:</b> <p>No response.</p>
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<pre>
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<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
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<span class="ansi-red-fg">NotImplementedError</span> Traceback (most recent call last)
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Input <span class="ansi-green-fg">In [11]</span>, in <span class="ansi-cyan-fg"><cell line: 5></span><span class="ansi-blue-fg">()</span>
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<span class="ansi-green-intense-fg ansi-bold"> 1</span> <span class="ansi-bold" style="color: rgb(0,135,0)">from</span> <span class="ansi-bold" style="color: rgb(0,0,255)">scipy</span><span class="ansi-bold" style="color: rgb(0,0,255)">.</span><span class="ansi-bold" style="color: rgb(0,0,255)">optimize</span> <span class="ansi-bold" style="color: rgb(0,135,0)">import</span> curve_fit
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<span class="ansi-green-intense-fg ansi-bold"> 3</span> <span style="color: rgb(95,135,135)"># Fitting</span>
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<span class="ansi-green-intense-fg ansi-bold"> 4</span> <span style="color: rgb(95,135,135)"># YOUR CODE HERE</span>
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<span class="ansi-green-fg">----> 5</span> <span class="ansi-bold" style="color: rgb(0,135,0)">raise</span> <span class="ansi-bold" style="color: rgb(215,95,95)">NotImplementedError</span>()
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<span class="ansi-green-intense-fg ansi-bold"> 6</span> <span style="color: rgb(0,135,0)">print</span>(<span style="color: rgb(175,0,0)">"</span><span style="color: rgb(175,0,0)">Fit parameters: a = </span><span class="ansi-bold" style="color: rgb(175,95,135)">{}</span><span style="color: rgb(175,0,0)">, beta = </span><span class="ansi-bold" style="color: rgb(175,95,135)">{}</span><span style="color: rgb(175,0,0)">"</span><span style="color: rgb(98,98,98)">.</span>format(a_fit,beta_fit))
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<span class="ansi-red-fg">NotImplementedError</span>: </pre>
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<div class="cell border-box-sizing code_cell rendered">
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<div class="prompt input_prompt">In [12]:</div><div class="panel panel-primary nbgrader_cell">
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<a name="comment-cell-7ebce3629edd06a2"></a><a name="cell-7ebce3629edd06a2"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right">
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Score: 0.0 / 5.0 <a href="#top">(Top)</a>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># Plotting</span>
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<span class="c1"># YOUR CODE HERE</span>
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<span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
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</pre></div>
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<b>Comments:</b> <p>No response.</p>
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<pre>
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<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
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<span class="ansi-red-fg">NotImplementedError</span> Traceback (most recent call last)
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Input <span class="ansi-green-fg">In [12]</span>, in <span class="ansi-cyan-fg"><cell line: 3></span><span class="ansi-blue-fg">()</span>
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<span class="ansi-green-intense-fg ansi-bold"> 1</span> <span style="color: rgb(95,135,135)"># Plotting</span>
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<span class="ansi-green-intense-fg ansi-bold"> 2</span> <span style="color: rgb(95,135,135)"># YOUR CODE HERE</span>
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<span class="ansi-green-fg">----> 3</span> <span class="ansi-bold" style="color: rgb(0,135,0)">raise</span> <span class="ansi-bold" style="color: rgb(215,95,95)">NotImplementedError</span>()
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<span class="ansi-red-fg">NotImplementedError</span>: </pre>
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<p><strong>(b)</strong> Produce a histogram of the height $H / \sqrt{n}$ for $n = 2^5, 2^6, \ldots, 2^{11}$ and $3000$ samples each and demonstrate with a plot that it appears to converge in distribution as $n\to\infty$. <em>Hint</em>: you could call <code>plt.hist(...,density=True,histtype='step')</code> for each $n$ to plot the densities on top of each other. <strong>(10 pts)</strong></p>
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<div class="cell border-box-sizing code_cell rendered">
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<div class="prompt input_prompt">In [13]:</div><div class="panel panel-primary nbgrader_cell">
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<a name="comment-cell-b2bb42d9970a7271"></a><a name="cell-b2bb42d9970a7271"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right">
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Score: 0.0 / 10.0 <a href="#top">(Top)</a>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="c1"># YOUR CODE HERE</span>
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<span class="k">raise</span> <span class="ne">NotImplementedError</span><span class="p">()</span>
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</pre></div>
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<b>Comments:</b> <p>No response.</p>
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<pre>
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<span class="ansi-red-fg">---------------------------------------------------------------------------</span>
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<span class="ansi-red-fg">NotImplementedError</span> Traceback (most recent call last)
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Input <span class="ansi-green-fg">In [13]</span>, in <span class="ansi-cyan-fg"><cell line: 2></span><span class="ansi-blue-fg">()</span>
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<span class="ansi-green-intense-fg ansi-bold"> 1</span> <span style="color: rgb(95,135,135)"># YOUR CODE HERE</span>
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<span class="ansi-green-fg">----> 2</span> <span class="ansi-bold" style="color: rgb(0,135,0)">raise</span> <span class="ansi-bold" style="color: rgb(215,95,95)">NotImplementedError</span>()
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<span class="ansi-red-fg">NotImplementedError</span>: </pre>
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