1153 lines
273 KiB
HTML
1153 lines
273 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_02</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_02 (Score: 99.0 / 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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<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=" 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 2</strong></p>
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<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>
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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="kn">from</span> <span class="nn">scipy.integrate</span> <span class="kn">import</span> <span class="n">quad</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">inversion_sample</span><span class="p">(</span><span class="n">f_inverse</span><span class="p">):</span>
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<span class="sd">'''Obtain an inversion sample based on the inverse-CDF f_inverse.'''</span>
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<span class="k">return</span> <span class="n">f_inverse</span><span class="p">(</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">def</span> <span class="nf">compare_plot</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span><span class="n">pdf</span><span class="p">,</span><span class="n">xmin</span><span class="p">,</span><span class="n">xmax</span><span class="p">,</span><span class="n">bins</span><span class="p">):</span>
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<span class="sd">'''Draw a plot comparing the histogram of the samples to the expectation coming from the pdf.'''</span>
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<span class="n">xval</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="n">xmin</span><span class="p">,</span><span class="n">xmax</span><span class="p">,</span><span class="n">bins</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
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<span class="n">binsize</span> <span class="o">=</span> <span class="p">(</span><span class="n">xmax</span><span class="o">-</span><span class="n">xmin</span><span class="p">)</span><span class="o">/</span><span class="n">bins</span>
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<span class="c1"># Calculate the expected numbers by numerical integration of the pdf over the bins</span>
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<span class="n">expected</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="n">quad</span><span class="p">(</span><span class="n">pdf</span><span class="p">,</span><span class="n">xval</span><span class="p">[</span><span class="n">i</span><span class="p">],</span><span class="n">xval</span><span class="p">[</span><span class="n">i</span><span class="o">+</span><span class="mi">1</span><span class="p">])[</span><span class="mi">0</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">bins</span><span class="p">)])</span><span class="o">/</span><span class="n">binsize</span>
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<span class="n">measured</span> <span class="o">=</span> <span class="n">np</span><span class="o">.</span><span class="n">histogram</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span><span class="n">bins</span><span class="p">,(</span><span class="n">xmin</span><span class="p">,</span><span class="n">xmax</span><span class="p">))[</span><span class="mi">0</span><span class="p">]</span><span class="o">/</span><span class="p">(</span><span class="nb">len</span><span class="p">(</span><span class="n">samples</span><span class="p">)</span><span class="o">*</span><span class="n">binsize</span><span class="p">)</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">xval</span><span class="p">,</span><span class="n">np</span><span class="o">.</span><span class="n">append</span><span class="p">(</span><span class="n">expected</span><span class="p">,</span><span class="n">expected</span><span class="p">[</span><span class="o">-</span><span class="mi">1</span><span class="p">]),</span><span class="s2">"-k"</span><span class="p">,</span><span class="n">drawstyle</span><span class="o">=</span><span class="s2">"steps-post"</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">bar</span><span class="p">((</span><span class="n">xval</span><span class="p">[:</span><span class="o">-</span><span class="mi">1</span><span class="p">]</span><span class="o">+</span><span class="n">xval</span><span class="p">[</span><span class="mi">1</span><span class="p">:])</span><span class="o">/</span><span class="mi">2</span><span class="p">,</span><span class="n">measured</span><span class="p">,</span><span class="n">width</span><span class="o">=</span><span class="n">binsize</span><span class="p">)</span>
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<span class="n">plt</span><span class="o">.</span><span class="n">xlim</span><span class="p">(</span><span class="n">xmin</span><span class="p">,</span><span class="n">xmax</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><span class="s2">"expected"</span><span class="p">,</span><span class="s2">"histogram"</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="k">def</span> <span class="nf">gaussian</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">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">x</span><span class="o">*</span><span class="n">x</span><span class="o">/</span><span class="mi">2</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="mi">2</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">pi</span><span class="p">)</span>
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</pre></div>
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<h2 id="Sampling-random-variables-via-the-inversion-method">Sampling random variables via the inversion method<a class="anchor-link" href="#Sampling-random-variables-via-the-inversion-method">¶</a></h2><p><strong>(35 Points)</strong></p>
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<p>Recall from the lecture that for any real random variable $X$ we can construct an explicit random variable via the inversion method that is identically distributed. This random variable is given by $F_X^{-1}(U)$ where $F_X$ is the CDF of $X$ and $U$ is a uniform random variable on $(0,1)$ and</p>
|
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$$
|
|
F_X^{-1}(p) := \inf\{ x\in\mathbb{R} : F_X(x) \geq p\}.
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$$<p>This gives a very general way of sampling $X$ in a computer program, as you will find out in this exercise.</p>
|
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<p><strong>(a)</strong> Let $X$ be an <strong>exponential random variable</strong> with <strong>rate</strong> $\lambda$, i.e. a continuous random variable with probability density function $f_X(x) = \lambda e^{-\lambda x}$ for $x > 0$. Write a function <code>f_inverse_exponential</code> that computes $F_X^{-1}(p)$. Illustrate the corresponding sampler with the help of the function <code>compare_plot</code> above. <strong>(10 pts)</strong></p>
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<a name="comment-cell-311fd25e116f5066"></a><a name="cell-311fd25e116f5066"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right">
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Score: 5.0 / 5.0 <a href="#top">(Top)</a>
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</span></div>
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<p>Reasoning from the PDF, we can find the CDF and invert that as follows.</p>
|
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$$
|
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f_X(x) = \lambda{}e^{-\lambda{}x}
|
|
$$$$
|
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\implies F_X(x)
|
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= \int_{-\infty}^x f_X(t)dt
|
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= \int_0^x \lambda{}e^{-\lambda{}t}dt
|
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= \left[ -e^{\lambda{}t} \right]_{t = 0}^x
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= 1 - e^{\lambda{}x}
|
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= \mathbb{P}(X \leq x) = p
|
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$$<p>for $x \in [0, \infty)$, otherwise zero.</p>
|
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<p>Now we seek $x$ as a function of $p$.</p>
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$$
|
|
1 - e^{\lambda{}x} = p
|
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\iff -\lambda{}x = \ln{(1-p)}
|
|
\iff x = \frac{\ln{(1-p)}}{-\lambda} = F^{-1}_X(p)
|
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$$<p>which works, as $1 - p \geq 0$ as $p \in [0, 1]$, allowing $\ln{0} = -\infty$.</p>
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<a name="comment-cell-06ef7d054d38f5c6"></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_inv_exponential</span><span class="p">(</span><span class="n">lam</span><span class="p">,</span><span class="n">p</span><span class="p">):</span>
|
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<span class="k">return</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="mi">1</span> <span class="o">-</span> <span class="n">p</span><span class="p">)</span><span class="o">/</span><span class="n">lam</span>
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<span class="n">f_X</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">,</span> <span class="n">lam</span><span class="p">:</span> <span class="n">lam</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">exp</span><span class="p">(</span><span class="o">-</span><span class="n">lam</span><span class="o">*</span><span class="n">x</span><span class="p">)</span> <span class="k">if</span> <span class="n">x</span> <span class="o">>=</span> <span class="mi">0</span> <span class="k">else</span> <span class="mi">0</span>
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<span class="c1"># Input parameters as list for flexibility in testing.</span>
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<span class="k">for</span> <span class="n">lam</span> <span class="ow">in</span> <span class="p">[</span><span class="mf">1.5</span><span class="p">]:</span>
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<span class="n">pdf</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">f_X</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">lam</span><span class="p">)</span>
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<span class="n">samples</span> <span class="o">=</span> <span class="p">[</span><span class="n">inversion_sample</span><span class="p">(</span><span class="k">lambda</span> <span class="n">p</span><span class="p">:</span> <span class="n">f_inv_exponential</span><span class="p">(</span><span class="n">lam</span><span class="p">,</span> <span class="n">p</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">100000</span><span class="p">)]</span>
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<span class="n">compare_plot</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">30</span><span class="p">)</span>
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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-2022e00546cf1bb0"></a><div class="panel-heading"><span class="nbgrader-label">Grade cell: <code>cell-2022e00546cf1bb0</code></span>
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Score: 5.0 / 5.0 <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="n">assert_almost_equal</span><span class="p">(</span><span class="n">f_inv_exponential</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span><span class="mf">0.6</span><span class="p">),</span><span class="mf">0.916</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.001</span><span class="p">)</span>
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<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">f_inv_exponential</span><span class="p">(</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.2</span><span class="p">),</span><span class="mf">0.743</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.001</span><span class="p">)</span>
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<span class="c1">### BEGIN HIDDEN TESTS</span>
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<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">f_inv_exponential</span><span class="p">(</span><span class="mf">4.3</span><span class="p">,</span><span class="mf">0.02</span><span class="p">),</span><span class="mf">0.0046983</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.0000001</span><span class="p">)</span>
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<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">f_inv_exponential</span><span class="p">(</span><span class="mf">0.2</span><span class="p">,</span><span class="mf">0.98</span><span class="p">),</span><span class="mf">19.560</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.001</span><span class="p">)</span>
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<span class="c1">### END HIDDEN TESTS</span>
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</pre></div>
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<p><strong>(b)</strong> Let now $X$ have the <strong>Pareto distribution</strong> of <strong>shape</strong> $\alpha > 0$ on $(b,\infty)$, which has probability density function $f_X(x) = \alpha b^{\alpha} x^{-\alpha-1}$ for $x > b$. Write a function <code>f_inv_pareto</code> that computes $F_X^{-1}(p)$. Compare a histogram with a plot of $f_X(x)$ to verify your function numerically. <strong>(10 pts)</strong></p>
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<a name="comment-cell-199713328dcd510d"></a><a name="cell-199713328dcd510d"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right">
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Score: 5.0 / 5.0 <a href="#top">(Top)</a>
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$$
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f_X(x) = \alpha b^{\alpha} x^{-\alpha-1}
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\\
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\implies F_X(x) = \int_{-\infty}^x f_X(t)dt
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= \int_{b}^x \alpha{}b^\alpha{}t^{-\alpha-1}dt
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= \alpha{}b^\alpha{} \int_{b}^x t^{-\alpha-1}dt
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= \alpha{}b^\alpha{} \left[ \frac{-t^{-\alpha}}{\alpha} \right]_{t = b}^x
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= b^\alpha (b^{-\alpha} - x^{-\alpha}) = 1 - b^\alpha x^{-\alpha} = p,
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$$<p>for $x > b$, otherwise $F_X(x) = 0$.</p>
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<p>To find $F_X^{-1}(p)$, we write $p$ as function of $x$.</p>
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$$
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p = 1 - b^\alpha x^{-\alpha}
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\iff b^\alpha x^{-\alpha} = 1 - p
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\iff x^{-\alpha} - b^{-\alpha}(1-p)
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\iff x = \frac{b}{(1-p)^{1/\alpha}}
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$$<p>Thus, $F_X^{-1}(p) = \frac{b}{(1-p)^{1/\alpha}}$ for $p \in [0, 1]$.</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-074f6a1fd6375c22"></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="c1">### Solution</span>
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<span class="k">def</span> <span class="nf">f_inv_pareto</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">p</span><span class="p">):</span>
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<span class="k">return</span> <span class="n">b</span><span class="o">/</span><span class="p">(</span><span class="mi">1</span><span class="o">-</span><span class="n">p</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="n">alpha</span><span class="p">)</span>
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<span class="c1"># plotting</span>
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<span class="n">f_X</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">:</span> <span class="n">alpha</span><span class="o">*</span><span class="n">b</span><span class="o">**</span><span class="n">alpha</span><span class="o">*</span><span class="n">x</span><span class="o">**</span><span class="p">(</span><span class="o">-</span><span class="n">alpha</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="k">if</span> <span class="n">x</span> <span class="o">>=</span> <span class="n">b</span> <span class="k">else</span> <span class="mi">0</span>
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<span class="c1"># Input parameters as list for flexibility in testing.</span>
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<span class="k">for</span> <span class="n">params</span> <span class="ow">in</span> <span class="p">[(</span><span class="mf">3.</span><span class="p">,</span> <span class="mf">1.</span><span class="p">)]:</span>
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<span class="n">alpha</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">params</span>
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<span class="n">pdf</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">f_X</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
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<span class="n">samples</span> <span class="o">=</span> <span class="p">[</span><span class="n">inversion_sample</span><span class="p">(</span><span class="k">lambda</span> <span class="n">p</span><span class="p">:</span> <span class="n">f_inv_pareto</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">p</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">100000</span><span class="p">)]</span>
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<span class="n">compare_plot</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="n">b</span><span class="o">-</span><span class="mi">1</span><span class="p">,</span> <span class="mi">4</span><span class="p">,</span> <span class="mi">30</span><span class="p">)</span>
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<img src="data:image/png;base64,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<div class="cell border-box-sizing code_cell rendered">
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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-726b321246679d28"></a><div class="panel-heading"><span class="nbgrader-label">Grade cell: <code>cell-726b321246679d28</code></span>
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<span class="pull-right">
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Score: 5.0 / 5.0 <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="n">assert_almost_equal</span><span class="p">(</span><span class="n">f_inv_pareto</span><span class="p">(</span><span class="mf">1.0</span><span class="p">,</span><span class="mf">1.5</span><span class="p">,</span><span class="mf">0.6</span><span class="p">),</span><span class="mf">3.75</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.0001</span><span class="p">)</span>
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<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">f_inv_pareto</span><span class="p">(</span><span class="mf">2.0</span><span class="p">,</span><span class="mf">2.25</span><span class="p">,</span><span class="mf">0.3</span><span class="p">),</span><span class="mf">2.689</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.001</span><span class="p">)</span>
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<span class="c1">### BEGIN HIDDEN TESTS</span>
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<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">f_inv_pareto</span><span class="p">(</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">3.5</span><span class="p">,</span><span class="mf">0.3</span><span class="p">),</span><span class="mf">123.90</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.1</span><span class="p">)</span>
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<span class="n">assert_almost_equal</span><span class="p">(</span><span class="n">f_inv_pareto</span><span class="p">(</span><span class="mf">3.0</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.3</span><span class="p">),</span><span class="mf">0.56312</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.0001</span><span class="p">)</span>
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<span class="c1">### END HIDDEN TESTS</span>
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</pre></div>
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<p><strong>(c)</strong> Let $X$ be a discrete random variable taking values in $\{1,2,\ldots,n\}$. Write a Python function <code>f_inv_discrete</code> that takes the probability mass function $p_X$ as a list <code>prob_list</code> given by $[p_X(1),\ldots,p_X(n)]$ and returns a random sample with the distribution of $X$ using the inversion method. Verify the working of your function numerically on an example. <strong>(15 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-694eb1261c2dc217"></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_inv_discrete</span><span class="p">(</span><span class="n">prob_list</span><span class="p">,</span><span class="n">p</span><span class="p">):</span>
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<span class="k">assert</span> <span class="n">np</span><span class="o">.</span><span class="n">isclose</span><span class="p">(</span><span class="n">np</span><span class="o">.</span><span class="n">sum</span><span class="p">(</span><span class="n">prob_list</span><span class="p">),</span> <span class="mi">1</span><span class="p">),</span> <span class="s2">"The probabilities should sum to one."</span>
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<span class="n">p_cum</span> <span class="o">=</span> <span class="mi">0</span>
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<span class="n">i</span> <span class="o">=</span> <span class="mi">0</span>
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<span class="k">while</span> <span class="n">p_cum</span> <span class="o"><</span> <span class="n">p</span><span class="p">:</span>
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<span class="n">p_cum</span> <span class="o">+=</span> <span class="n">prob_list</span><span class="p">[</span><span class="n">i</span><span class="p">]</span>
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<span class="n">i</span> <span class="o">+=</span> <span class="mi">1</span>
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<span class="k">return</span> <span class="n">i</span>
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<span class="c1"># plotting</span>
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<span class="n">f_X</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">prob_list</span><span class="p">,</span> <span class="n">x</span><span class="p">:</span> <span class="n">prob_list</span><span class="p">[</span><span class="n">np</span><span class="o">.</span><span class="n">rint</span><span class="p">(</span><span class="n">x</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> <span class="o">-</span> <span class="mi">1</span><span class="p">]</span> <span class="k">if</span> <span class="n">np</span><span class="o">.</span><span class="n">rint</span><span class="p">(</span><span class="n">x</span><span class="p">)</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="nb">len</span><span class="p">(</span><span class="n">prob_list</span><span class="p">)</span> <span class="o">+</span> <span class="mi">1</span><span class="p">)</span> <span class="k">else</span> <span class="mi">0</span>
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<span class="c1"># Input parameters as list for flexibility in testing.</span>
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<span class="k">for</span> <span class="n">prob_list</span> <span class="ow">in</span> <span class="p">[[</span><span class="mf">.1</span><span class="p">,</span> <span class="mf">.3</span><span class="p">,</span> <span class="mf">.2</span><span class="p">,</span> <span class="mf">.4</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.5</span><span class="p">,</span> <span class="mf">0.5</span><span class="p">],</span> <span class="p">[</span><span class="mf">0.7</span><span class="p">,</span> <span class="mf">0.1</span><span class="p">,</span> <span class="mf">0.2</span><span class="p">]]:</span>
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<span class="n">alpha</span><span class="p">,</span> <span class="n">b</span> <span class="o">=</span> <span class="n">params</span>
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<span class="n">pdf</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">f_X</span><span class="p">(</span><span class="n">prob_list</span><span class="p">,</span> <span class="n">x</span><span class="p">)</span>
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<span class="n">samples</span> <span class="o">=</span> <span class="p">[</span><span class="n">inversion_sample</span><span class="p">(</span><span class="k">lambda</span> <span class="n">p</span><span class="p">:</span> <span class="n">f_inv_discrete</span><span class="p">(</span><span class="n">prob_list</span><span class="p">,</span> <span class="n">p</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">100000</span><span class="p">)]</span>
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<span class="n">compare_plot</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="mf">.5</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">prob_list</span><span class="p">)</span> <span class="o">+</span> <span class="mf">.5</span><span class="p">,</span> <span class="nb">len</span><span class="p">(</span><span class="n">prob_list</span><span class="p">))</span>
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src="data:image/png;base64,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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="cell-140af6b31464fbef"></a><div class="panel-heading"><span class="nbgrader-label">Grade cell: <code>cell-140af6b31464fbef</code></span>
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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_inv_discrete</span><span class="p">([</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.5</span><span class="p">],</span><span class="mf">0.4</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_inv_discrete</span><span class="p">([</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.5</span><span class="p">],</span><span class="mf">0.8</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_inv_discrete</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mi">1</span><span class="p">],</span><span class="mf">0.1</span><span class="p">)</span><span class="o">==</span><span class="mi">3</span>
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<span class="c1">### BEGIN HIDDEN TESTS</span>
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<span class="k">assert</span> <span class="n">f_inv_discrete</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span><span class="mf">0.42</span><span class="p">)</span><span class="o">==</span><span class="mi">5</span>
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<span class="k">assert</span> <span class="n">f_inv_discrete</span><span class="p">([</span><span class="mi">0</span><span class="p">,</span><span class="mi">0</span><span class="p">,</span><span class="mf">0.1</span><span class="p">,</span><span class="mf">0.3</span><span class="p">,</span><span class="mf">0.5</span><span class="p">,</span><span class="mf">0.1</span><span class="p">,</span><span class="mi">0</span><span class="p">],</span><span class="mf">0.02</span><span class="p">)</span><span class="o">==</span><span class="mi">3</span>
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<span class="c1">### END HIDDEN TESTS</span>
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<h2 id="Central-limit-theorem?">Central limit theorem?<a class="anchor-link" href="#Central-limit-theorem?">¶</a></h2><p><strong>(35 Points)</strong></p>
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<p>In this exercise we will have a closer look at central limits of the Pareto distribution, for which you implemented a random sampler in the previous exercise. By performing the appropriate integrals it is straightforward to show that</p>
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$$
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\mathbb{E}[X] = \begin{cases} \infty & \text{for }\alpha \leq 1 \\ \frac{\alpha b}{\alpha - 1} & \text{for }\alpha > 1 \end{cases}, \qquad \operatorname{Var}(X) = \begin{cases} \infty & \text{for }\alpha \leq 2 \\ \frac{\alpha b^2}{(\alpha - 1)^2(\alpha-2)} & \text{for }\alpha > 2 \end{cases}.
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$$<p>This shows in particular that the distribution is <strong>heavy tailed</strong>, in the sense that some moments $\mathbb{E}[X^k]$ diverge.</p>
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<p><strong>(a)</strong> Write a function <code>sample_Zn</code> that produces a random sample for $Z_n= \frac{\sqrt{n}}{\sigma_X}(\bar{X}_n - \mathbb{E}[X])$ given $\alpha>2$, $b>0$ and $n\geq 1$. Visually verify the central limit theorem for $\alpha = 4$, $b=1$ and $n=1000$ by comparing a histogram of $Z_n$ to the standard normal distribution (you may use <code>compare_plot</code>). <strong>(10 pts)</strong></p>
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<div class="prompt input_prompt">In [9]:</div><div class="panel panel-primary nbgrader_cell">
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<a name="comment-cell-b7186322b09717f8"></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">sample_Zn</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span><span class="n">b</span><span class="p">,</span><span class="n">n</span><span class="p">):</span>
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<span class="k">assert</span> <span class="n">alpha</span> <span class="o">></span> <span class="mi">2</span>
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<span class="k">assert</span> <span class="n">b</span> <span class="o">></span> <span class="mi">0</span>
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<span class="k">assert</span> <span class="n">n</span> <span class="o">>=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">type</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="nb">int</span>
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<span class="n">E_X</span> <span class="o">=</span> <span class="n">alpha</span><span class="o">*</span><span class="n">b</span><span class="o">/</span><span class="p">(</span><span class="n">alpha</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
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<span class="n">Var_X</span> <span class="o">=</span> <span class="n">alpha</span><span class="o">*</span><span class="n">b</span><span class="o">**</span><span class="mi">2</span><span class="o">/</span><span class="p">(</span> <span class="p">(</span><span class="n">alpha</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span><span class="o">**</span><span class="mi">2</span><span class="o">*</span><span class="p">(</span><span class="n">alpha</span> <span class="o">-</span> <span class="mi">2</span><span class="p">)</span> <span class="p">)</span>
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<span class="n">inv_pareto_samples</span> <span class="o">=</span> <span class="p">[</span><span class="n">inversion_sample</span><span class="p">(</span><span class="k">lambda</span> <span class="n">p</span><span class="p">:</span> <span class="n">f_inv_pareto</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">p</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">sqrt</span><span class="p">(</span><span class="n">n</span><span class="o">/</span><span class="n">Var_X</span><span class="p">)</span><span class="o">*</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">inv_pareto_samples</span><span class="p">)</span> <span class="o">-</span> <span class="n">E_X</span><span class="p">)</span>
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<span class="c1"># Plotting</span>
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<span class="n">alpha</span> <span class="o">=</span> <span class="mi">4</span>
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<span class="n">b</span> <span class="o">=</span> <span class="mi">1</span>
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<span class="n">n</span> <span class="o">=</span> <span class="mi">1000</span>
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<span class="n">pdf</span> <span class="o">=</span> <span class="n">gaussian</span>
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<span class="n">samples</span> <span class="o">=</span> <span class="p">[</span><span class="n">inversion_sample</span><span class="p">(</span><span class="k">lambda</span> <span class="n">p</span><span class="p">:</span> <span class="n">sample_Zn</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">n</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">1000</span><span class="p">)]</span>
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<span class="n">compare_plot</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
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</pre></div>
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<b>Comments:</b> <p>Use less bins to prevent overfitting</p>
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<img src="data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAXQAAAD4CAYAAAD8Zh1EAAAAOXRFWHRTb2Z0d2FyZQBNYXRwbG90bGliIHZlcnNpb24zLjUuMSwgaHR0cHM6Ly9tYXRwbG90bGliLm9yZy/YYfK9AAAACXBIWXMAAAsTAAALEwEAmpwYAAAca0lEQVR4nO3df3BU9b3/8ecbBFNT/FGF2hGR0K+WRpKe2hAFHdkK2tSqXKd1wNj6o1dXbP01U/1KvZaZWkdxdFp/3i/NtFZvayp38EcZ5arXH9Fqpfyw2yixWApYo1IsVgwIIvL+/rHJcrJskiXZzdmcfT1mHPfsOdm8OcY377zP54e5OyIiMvQNizoAEREpDCV0EZGYUEIXEYkJJXQRkZhQQhcRiYl9ovrGhxxyiI8fPz6qby8iMiStXLnyn+4+Ote5yBL6+PHjWbFiRVTfXkRkSDKzN3o6p5aLiEhMKKGLiMSEErqISExE1kMXkXj4+OOPaW9vZ/v27VGHEisVFRWMHTuWESNG5P01SugiMiDt7e2MGjWK8ePHY2ZRhxML7s6mTZtob2+nqqoq769Ty0VEBmT79u0cfPDBSuYFZGYcfPDBe/1bjxK6iAyYknnh9eeeKqGLiMSEErqISD+kUimWLFmy11+XSCSKNqlSD0WlrIyf+1jm9fr534gwEhnqUqkUK1as4NRTT406lIy8KnQzazCz1Wa2xszm5jifMLPNZpbq/Gde4UMVEenZb37zG+rr6wmCgIsvvpg//vGP1NbWsn37drZu3crRRx/Nq6++SktLCyeeeCJnnnkm1dXVzJkzh127dgHw5JNPMmXKFI455hjOOusstmzZAsDy5cuZOnUqX/rSl6ivr2fz5s3MmzePhQsXEgQBCxcuZOvWrXz3u99l8uTJfPnLX+Z3v/sdANu2bWP27NnU1tYya9Ystm3bVrR70GeFbmbDgbuBk4F2YLmZLXb3tqxLf+/upxUhRhEZIq688kpSqVRBPzMIAm677bZer3nttddYuHAhL774IiNGjOB73/seq1ev5owzzuC6665j27ZtfPvb32bSpEm0tLSwbNky2traOOKII2hoaOChhx4ikUhwww038NRTT1FZWcnNN9/MT3/6U+bOncusWbNYuHAhkydP5oMPPmC//fbj+uuvZ8WKFdx1110AXHvttZx00kncc889vP/++9TX1zNjxgx+/vOfs99++9Ha2kprayvHHHNMQe9PWD4tl3pgjbuvBTCzB4CZQHZCFxGJxNNPP83KlSuZPHkykK6Kx4wZw7x585g8eTIVFRXccccdmevr6+uZMGECAGeffTYvvPACFRUVtLW1cfzxxwOwY8cOpkyZwurVq/nc5z6X+ez9998/ZwxPPvkkixcv5tZbbwXSwzn//ve/8/zzz3P55ZcDUFtbS21tbXFuAvkl9MOAN0PH7cCxOa6bYmZ/Bt4GrnL3VdkXmFkSSAKMGzdu76MVkZLWVyVdLO7Oeeedx0033dTt/Q0bNrBlyxY+/vhjtm/fTmVlJbDnkEAzw905+eST+e1vf9vtXGtra15DCN2dBx98kC984Qt7nBusYZ359NBzReJZxy8DR7j7l4A7gUdyfZC7N7l7nbvXjR6dczlfEZG9Nn36dBYtWsTGjRsBeO+993jjjTdIJpP85Cc/4ZxzzuGaa67JXL9s2TLWrVvHrl27WLhwISeccALHHXccL774ImvWrAHgww8/5PXXX2fixIm8/fbbLF++HICOjg527tzJqFGj6OjoyHzm1772Ne68807c0+nxT3/6EwAnnngi999/PwCvvvoqra2tRbsP+VTo7cDhoeOxpKvwDHf/IPR6iZn9p5kd4u7/LEyYIiI9q66u5oYbbuCUU05h165djBgxgpkzZ7LPPvvQ2NjIJ598wtSpU3nmmWcYNmwYU6ZMYe7cubzyyiuZB6TDhg3j3nvv5eyzz+ajjz4C4IYbbuCoo45i4cKFXHbZZWzbto1PfepTPPXUU3z1q19l/vz5BEHAD3/4Q370ox9x5ZVXUltbi7szfvx4Hn30US655BIuuOACamtrCYKA+vr6ot0H6/rbpMcLzPYBXgemA28By4HGcEvFzA4F/uHubmb1wCLSFXuPH15XV+fa4EIGm4YtFt5rr73GF7/4xajDyFtLSwu33norjz76aNSh9CnXvTWzle5el+v6Pit0d99pZpcCTwDDgXvcfZWZzek8vwD4FnCJme0EtgGze0vmIiJSeHlNLHL3JcCSrPcWhF7fBdxV2NBERAovkUiQSCSiDqMoNPVfRCQmlNBFRGJCCV1EJCaU0EVEYkKrLYpIQYWHhhZCPsNL169fz2mnncarr77a7f158+Zx4oknMmPGjJxf98gjj3DUUUdRXV1dkFijpgpdRGLr+uuv7zGZQzqht7UVZlmqnTt3FuRzBkIJXWJn/NzHMv9I+fjkk0+46KKLOProoznllFPYtm0b559/PosWLQJg7ty5VFdXU1tby1VXXcUf/vAHFi9ezNVXX00QBPztb38jlUpx3HHHUVtby5lnnsm//vUvIL18bm1tLVOmTOHqq69m0qRJANx7772cddZZnH766Zxyyils2bKF6dOnc8wxx1BTU5NZQnf9+vVMnDiRCy+8kEmTJnHOOefw1FNPcfzxx3PkkUeybNmygtwDJXQRiYW//vWvfP/732fVqlUceOCBPPjgg5lz7733Hg8//DCrVq2itbWV6667jqlTp3LGGWdwyy23kEql+PznP8+5557LzTffTGtrKzU1Nfz4xz8G4IILLmDBggW89NJLDB8+vNv3femll7jvvvt45plnqKio4OGHH+bll1/m2Wef5Qc/+EFmbZc1a9ZwxRVX0Nrayl/+8heam5t54YUXuPXWW7nxxhsLcg+U0EUkFqqqqgiCAICvfOUrrF+/PnNu//33p6KiggsvvJCHHnqI/fbbb4+v37x5M++//z7Tpk0D4LzzzuP555/n/fffp6Ojg6lTpwLQ2NjY7etOPvlkPvOZzwDpFRevvfZaamtrmTFjBm+99Rb/+Mc/MvHV1NQwbNgwjj76aKZPn46ZUVNT0y3WgVBCF5FY2HfffTOvhw8f3q2nvc8++7Bs2TK++c1v8sgjj9DQ0JD35/a1iknXkrwA999/P++++y4rV64klUrx2c9+lu3bt+8R37BhwzLHw4YNK1j/XQldRGJvy5YtbN68mVNPPZXbbrsts6tSeAncAw44gIMOOojf//73APz6179m2rRpHHTQQYwaNYqlS5cC8MADD/T4fTZv3syYMWMYMWIEzz77LG+88UZx/2BZNGxRRAqqFFex7OjoYObMmWzfvh1352c/+xkAs2fP5qKLLuKOO+5g0aJF3HfffcyZM4cPP/yQCRMm8Ktf/QqAX/7yl1x00UVUVlaSSCQ44IADcn6fc845h9NPP526ujqCIGDixImD9meEPJbPLRYtnyvF0tsSuVo+t/CG2vK5/bFlyxY+/elPAzB//nzeeecdbr/99qJ/34IvnysiUu4ee+wxbrrpJnbu3MkRRxzBvffeG3VIOSmhi4j0YdasWcyaNSvqMPqkh6IiMmDaz6bw+nNPldBFZEAqKirYtGmTknoBuTubNm2ioqJir75OLRcRGZCxY8fS3t7Ou+++G3UosVJRUcHYsWP36muU0EX6oJExvRsxYgRVVVVRhyGo5SIiEhtK6CIiMaGELiISE+qhS6xpTXQpJ6rQRURiQgldRCQmlNBFRGJCCV1EJCaU0EVEYkIJXUQkJpTQRURiQgldRCQm8kroZtZgZqvNbI2Zze3luslm9omZfatwIYqISD76TOhmNhy4G/g6UA2cbWbVPVx3M/BEoYMUEZG+5VOh1wNr3H2tu+8AHgBm5rjuMuBBYGMB4xMRkTzls5bLYcCboeN24NjwBWZ2GHAmcBIwuacPMrMkkAQYN27c3sYqUhRNTU00Nzd3e2/p2k2Z15XVCUYFDTm/VmulSynJp0K3HO9l7zV1G3CNu3/S2we5e5O717l73ejRo/MMUaS4mpubSaVSOc/t2LiOrW0tgxqPSH/lU6G3A4eHjscCb2ddUwc8YGYAhwCnmtlOd3+kEEGKFFpH6nESiVsASKVSBEFAS0tL5nxX5b2heS47Nq5jQ3N6LEDThLdIJpODHq9IPvKp0JcDR5pZlZmNBGYDi8MXuHuVu4939/HAIuB7SuZSyra2tWSq8iAIaGxszHldZXWCkWPS26vt2Lhuj9aMSCnps0J3951mdinp0SvDgXvcfZWZzek8v6DIMYoURXZVnsuooCHTP9/QPJelazdpjXUpWXltcOHuS4AlWe/lTOTufv7AwxIpvI7U45l++I6N62DCwXv9GeH2C/T+wFRksGmmqJSNrW0t6UQOjBxT1WObpSfh9gvogamUHm1BJ2Vl5JgqDm2cD0AyuXfDDMPtF6BbpS5SClShi4jEhCp0ia1wzxzSLZJwy6QQk4I0pFFKiSp0ia1wzxzS7ZbK6kTBPl9DGqXUqEKXWAv3zAste0ijSNRUoYuIxIQSuohITCihi4jEhHroEitNTU1saL4T2HNUS7GlUikSiQQAjY2NGvEig04VusRKc3Nzt9mghRzV0pvK6gRBEADpxK4RLxIFVegSC5nlbtduKurIlp6MChpomZ/+zaCrShcZbKrQRURiQgldRCQmlNBFRGJCCV1EJCaU0EVEYkKjXGTIi3LseU/CY9JB49JlcCihy5DXNfZ85JiqQR173pPsnZC6NqNWQpdiU0KXWBjo2PNCbvycTCa7JW+NS5fBoh66iEhMKKGLiMSEErqISEyohy5DUrjnvWHtpggjESkdqtBFRGJCFboMOeFx51A6Y897o7XSZTAoocuQEx53DoO77nl/hMela0y6FJMSugxJUax53l/hcekaky7FpB66iEhMKKGLiMSEErqISEyohy4lK3t9lfXzvxFRJCJDQ14Vupk1mNlqM1tjZnNznJ9pZq1mljKzFWZ2QuFDFRGR3vRZoZvZcOBu4GSgHVhuZovdvS102dPAYnd3M6sF/huYWIyARUQkt3wq9HpgjbuvdfcdwAPAzPAF7r7F3b3zsBJwRAqoI/U4iUSCRCKRGcs9VHVNMkokEjQ1NUUdjsRIPj30w4A3Q8ftwLHZF5nZmcBNwBggZ7PTzJJAEmDcuHF7G6uUsa1tLaQ+eJMgCAiCgFcqaqIOaQ/hnn9P/X5NMpJiyqdCtxzv7VGBu/vD7j4R+DfgJ7k+yN2b3L3O3etGjx69V4GKBEFAS0sLLS0tjAoaog6nX5LJZObPEARB1OFIzOST0NuBw0PHY4G3e7rY3Z8HPm9mhwwwNhER2Qv5JPTlwJFmVmVmI4HZwOLwBWb2f8zMOl8fA4wEtKapiMgg6rOH7u47zexS4AlgOHCPu68yszmd5xcA3wTONbOPgW3ArNBDUpGCWLp2U0H3/hSJm7wmFrn7EmBJ1nsLQq9vBm4ubGgiIrI3NPVfRCQmlNBFRGJCa7lISRqKuxL1R3gnI9BuRjIwSuhSkobarkT9EZ5kBJpoJAOnhC4layjtStQf4Z2MQLsZycCphy4iEhNK6CIiMaGELiISE0roIiIxoYQuIhITGuUiJaVrrZYNa+O7tls+66YPxmdI/KhCFxGJCVXoUjLCs0PjOjO0L+GZo5o1KntLCV1KRnh2aBxnhvZF29PJQCmhS9H0p88b99mhvQnPHNWsUekP9dBFRGJCCV1EJCaU0EVEYkIJXUQkJpTQRURiQgldRCQmlNAlUk1NTSQSCRKJRGbstaR1TTJKJBI0NTVFHY4MARqHLpFqbm4mlUoRBAFBEPBKRU3UIRVFeEx+b+93jdfXJCPpDyV0iVwQBLS0tAA9J75yo0lG0h9quYiIxIQSuohITKjlIlIEah1JFFShi4jEhBK6iEhMKKGLiMSEErqISEzk9VDUzBqA24HhwC/cfX7W+XOAazoPtwCXuPufCxmoxENTUxPNzc2Z465JRdK78NZ0AB0VNYwKGqILSEpSnxW6mQ0H7ga+DlQDZ5tZddZl64Bp7l4L/ATQPGXJqWtmaJcgCLrNipQ9NTY2dvtLL5VKsbWtJbJ4pHTlU6HXA2vcfS2AmT0AzATaui5w9z+Erl8KjC1kkBIv4Zmh0rfwrFFIzxxdunZThBFJqcqnh34Y8GbouL3zvZ78O/A/AwlKRET2Xj4VuuV4z3NeaPZV0gn9hB7OJ4EkwLhx4/IMUUpZfzaC7u0zZLdC3FspL/lU6O3A4aHjscDb2ReZWS3wC2Cmu+f8fdDdm9y9zt3rRo8e3Z94RUSkB/kk9OXAkWZWZWYjgdnA4vAFZjYOeAj4jru/XvgwRUSkL322XNx9p5ldCjxBetjiPe6+yszmdJ5fAMwDDgb+08wAdrp7XfHCFhGRbHmNQ3f3JcCSrPcWhF5fCFxY2NAkTrr6wRvWbuK4CQdHHE3p0vMEGQjNFBURiQktnytF15F6PDMRZsfGdaAKfcB2bFzHhua5ADRNeEvb0wmgCl0Gwda2lnQiB0aOqdLM0AFqbGxk5JgqIJ3Yw0spSHlThS6DYuSYKg5tTC8BlExqTPVAJJNJblybntvXVaWLgCp0EZHYUEIXEYkJJXQRkZhQD11kCND4dMmHKnQRkZhQQhcRiQm1XKQompqa2NB8J5AeK901bloKL7w9XWNjoyYZlTEldCmK5ubmTCIfOaaKyupE1CHFUmV1gprtrwDpxL507abMGPVsWlM9/pTQpWjCk4mkOEYFDbTMT/8mpK3pRD10EZGYUEIXEYkJtVxk0GlMtUhxqEIXEYkJJXQRkZhQy0UKoiP1OInELZnjVCoF+x8eXUBlKrzxBaSHNY4KGiKMSAaTEroUxNa2FlIfvEkQBAAEQcArFTXRBlUmup5JdFTUMHLM7mGLXZuKKKGXDyV0KZggCGhpackc6+Hn4BoVNHRL3tr8ovyohy4iEhNK6CIiMaGWi0iZyG6BaW2X+FGFLiISE0roIiIxoZaL9Fv2mudMODjiiCRbeFy6xqTHnyp06beuNc8hvVRuY2NjxBFJWGV1IrOxyI6N69ja1hJtQFJ0qtBlQMJrnieTeshWSsLj0jUmvTyoQhcRiQkldBGRmFBCFxGJibwSupk1mNlqM1tjZns048xsopm9ZGYfmdlVhQ9TRET60udDUTMbDtwNnAy0A8vNbLG7t4Uuew+4HPi3YgQppaOpqYnm5mZAS+QONdlL6zZNeItkMhlhRFJo+VTo9cAad1/r7juAB4CZ4QvcfaO7Lwc+LkKMUkKam5vTiZz06oqV1YlI45H8hIcwQjq5d/3FLPGRz7DFw4A3Q8ftwLH9+WZmlgSSAOPGjevPR0gJCC+TqyVyh4a+ltYN/3fUGi9DVz4VuuV4z/vzzdy9yd3r3L1u9OjR/fkIERHpQT4JvR0IN0rHAm8XJxwREemvfBL6cuBIM6sys5HAbGBxccMSEZG91WcP3d13mtmlwBPAcOAed19lZnM6zy8ws0OBFcD+wC4zuxKodvcPihe6DJZwf3XD2k17vJfrOhEZfHmt5eLuS4AlWe8tCL3eQLoVIzETXlER0qMjwqMlZOhKpVIkEgkgvcG0VmIc+jRTVHoVXlER0otxaaji0FdZnSAIAiCd2LUSYzxotUXpU3hFRYmHUUEDLfPTv3klEgmWdrbSZGhThS4iEhNK6CIiMaGELiISE+qhyx60AFf5CS/cpUW7hi5V6LIHLcBVHsbPfYzxcx/jlYqabnuPatGuoUsVuuSkBbjKh/YejQ9V6CIiMaGELiISE2q5CJB+EHr5DemJJl3T+9VqKU/hJQEAGhsb9ZB0iFCFLkD3Kf6a3l++wksCQDq56yHp0KEKXTI0xV/CSwIA3Sp1KX2q0EVEYkIVepkKTx4CTSCS3bLXv9+xcR0V42oAuOO6y/Lqp2uP0mioQi9T4clDoAlEkltldUKTjoYQVehlLDx5CDSBSPakSUdDiyp0EZGYUEIvI01NTSQSCRKJRLd2i0i+usaoJxIJmpqaog5HsqjlUka6+uZBEBAEAa9U1KjNInmrrE6wta2FpZ0PSpeu3cSNaw+LOiwJUUIvM1p0S/pL/fTSp4Qec9lrm4dnAYoMRHgNdUhX8F0JX6KhHnrMZa9t3tjYGG1AEgvh4YyQTu5b21qiC0gAVehlIbvNcqNaLTJA4fYL9L8FowlIhaWEHjO5ZoCqzSKDIdyCUfslGmq5xEyuGaBqs0ixZc8oVfslGqrQYyDXg8/wDFCRYsseAaNNp6OhCj0G9OBTSkkx1n/p2tBaQ217pwp9iFJVLqUqu1rXDkiDRwl9iMh+2Pncc88BMG3aNFXlUrIqqxPUbH8lc/zcc8/x3HPPZX6WOypq9PC0gJTQS1g4iYcTeNe/VelIqcveAWnPn+nnMg9Q1WsfOCX0EhDuC3akHs/8gH/05qsA7Hv4JPY9fNIemwv0NKY8ezyv+o5SKpLJZGb9l89U1HT7Wb/44ou7/Rban4Il33Ht/R3/Xurj5vNK6GbWANwODAd+4e7zs85b5/lTgQ+B89395QLHGhvZ7ZMNazdlXmcn8fB43mSy9H6ARPor3GvvSD3ea2sm/P+IKvme9ZnQzWw4cDdwMtAOLDezxe7eFrrs68CRnf8cC/y/zn/HWnZizld2+yQsO4mLlIPeWjNhuSr5sHDiTyy9pdu5cmhR5lOh1wNr3H0tgJk9AMwEwgl9JvBf7u7AUjM70Mw+5+7v9PShq1evHvI7iveWmHuT3f9WS0Sku2QyuUd7Efas5PPVW8Wfnfh709+vGyyWzsG9XGD2LaDB3S/sPP4OcKy7Xxq65lFgvru/0Hn8NHCNu6/I+qwk0PVf6QvA6kL9QQbgEOCfUQdRInQvdtO92E33YrdSuBdHuPvoXCfyqdAtx3vZfwvkcw3u3gSU1DYnZrbC3euijqMU6F7spnuxm+7FbqV+L/KZKdoOHB46Hgu83Y9rRESkiPJJ6MuBI82sysxGArOBxVnXLAbOtbTjgM299c9FRKTw+my5uPtOM7sUeIL0sMV73H2Vmc3pPL8AWEJ6yOIa0sMWLyheyAVXUi2giOle7KZ7sZvuxW4lfS/6fCgqIiJDg1ZbFBGJCSV0EZGYUELvZGZXmZmb2SFRxxIVM7vFzP5iZq1m9rCZHRh1TIPNzBrMbLWZrTGz/m2UGQNmdriZPWtmr5nZKjO7IuqYomZmw83sT53zbkqSEjrpH17SSxv8PepYIva/wCR3rwVeB34YcTyDKrTMxdeBauBsM6uONqrI7AR+4O5fBI4Dvl/G96LLFcBrUQfRGyX0tJ8B/5cck6HKibs/6e47Ow+Xkp5PUE4yy1y4+w6ga5mLsuPu73QtsOfuHaQT2WHRRhUdMxsLfAP4RdSx9KbsE7qZnQG85e5/jjqWEvNd4H+iDmKQHQa8GTpup4yTWBczGw98GfhjxKFE6TbSRd+uiOPoVVmsh25mTwGH5jj1H8C1wCmDG1F0ersX7v67zmv+g/Sv3PcPZmwlIK8lLMqJmX0aeBC40t0/iDqeKJjZacBGd19pZomIw+lVWSR0d5+R630zqwGqgD+nl3RnLPCymdW7+4ZBDHHQ9HQvupjZecBpwHQvv0kKWsIixMxGkE7m97v7Q1HHE6HjgTPM7FSgAtjfzH7j7t+OOK49aGJRiJmtB+rcPerV1CLRuZHJT4Fp7v5u1PEMNjPbh/TD4OnAW6SXvWh091WRBhaBzk1r7gPec/crIw6nZHRW6Fe5+2kRh5JT2ffQpZu7gFHA/5pZyswWRB3QYOp8INy1zMVrwH+XYzLvdDzwHeCkzp+FVGeFKiVMFbqISEyoQhcRiQkldBGRmFBCFxGJCSV0EZGYUEIXEYkJJXQRkZhQQhcRiYn/DyU0N86dY4k3AAAAAElFTkSuQmCC
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</div>
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<div class="cell border-box-sizing code_cell rendered">
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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="cell-5d16b014bef9d86f"></a><div class="panel-heading"><span class="nbgrader-label">Grade cell: <code>cell-5d16b014bef9d86f</code></span>
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Score: 9.0 / 10.0 <a href="#top">(Top)</a>
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<div class=" highlight hl-ipython3"><pre><span></span><span class="n">assert_almost_equal</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">sample_Zn</span><span class="p">(</span><span class="mf">3.5</span><span class="p">,</span><span class="mf">2.1</span><span class="p">,</span><span class="mi">100</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">100</span><span class="p">)]),</span><span class="mi">0</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
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<span class="n">assert_almost_equal</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">sample_Zn</span><span class="p">(</span><span class="mf">3.5</span><span class="p">,</span><span class="mf">2.1</span><span class="p">,</span><span class="mi">100</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">100</span><span class="p">)]),</span><span class="mi">1</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
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<span class="c1">### BEGIN HIDDEN TESTS</span>
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<span class="n">assert_almost_equal</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">sample_Zn</span><span class="p">(</span><span class="mf">7.5</span><span class="p">,</span><span class="mf">0.7</span><span class="p">,</span><span class="mi">100</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">100</span><span class="p">)]),</span><span class="mi">0</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.3</span><span class="p">)</span>
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<span class="n">assert_almost_equal</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">sample_Zn</span><span class="p">(</span><span class="mf">7.5</span><span class="p">,</span><span class="mf">0.7</span><span class="p">,</span><span class="mi">100</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">100</span><span class="p">)]),</span><span class="mi">1</span><span class="p">,</span><span class="n">delta</span><span class="o">=</span><span class="mf">0.25</span><span class="p">)</span>
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<span class="c1">### END HIDDEN TESTS</span>
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</pre></div>
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<p><strong>(b)</strong> Now take $\alpha = 3/2$ and $b=1$.
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With some work (which you do not have to do) one can show that the characteristic function of $X$ admits the following expansion around $t=0$,</p>
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$$
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\varphi_X(t) = 1 + 3 i t - (|t|+i t)\,\sqrt{2\pi|t|} + O(t^{2}).
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$$<p>Based on this, prove the <strong>generalized CLT</strong> for this particular distribution $X$ which states that $Z_n = c\, n^{1/3} (\bar{X}_n - \mathbb{E}[X])$ in the limit $n\rightarrow\infty$ converges in distribution, with a to-be-determined choice of overall constant $c$, to a limiting random variable $\mathcal{S}$ with characteristic function</p>
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$$
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\varphi_{\mathcal{S}}(t) = \exp\big(-(|t|+it)\sqrt{|t|}\big).
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$$<p><strong>(15 pts)</strong></p>
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<a name="comment-cell-b25551eca32c4807"></a><a name="cell-b25551eca32c4807"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right">
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Score: 15.0 / 15.0 <a href="#top">(Top)</a>
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\begin{align}
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\phi_{Z_n}(t) &= \mathbb{E}\left[ e^{itZ_n} \right]
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\\ &= \mathbb{E}\left[ e^{itcn^{1/3}(\bar{X_n} - \mathbb{E}[X])} \right]
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\\ &= \mathbb{E}\left[ e^{itcn^{1/3}(\frac{1}{n}\sum_{i=1}^n X_i - \mathbb{E}[X])} \right]
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\\ &= \mathbb{E}\left[ \left( \prod_{i=1}^n e^{itcn^{-2/3}X_i} \right) e^{itcn^{1/3}\mathbb{E}[X])} \right]
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\\ &= \left( \prod_{i=1}^n \mathbb{E}\left[ e^{itcn^{-2/3}X_i} \right] \right)\mathbb{E}\left[ e^{itcn^{1/3}\mathbb{E}[X])} \right]
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\\ &= \left( \prod_{i=1}^n \phi_X(cn^{-2/3}t) \right)\mathbb{E}\left[ e^{itcn^{1/3}\mathbb{E}[X])} \right]
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\\ &= \left( \phi_X(cn^{-2/3}t) \right)^n \mathbb{E}\left[ e^{itcn^{1/3}\mathbb{E}[X])} \right]
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\end{align}<p>where we used the identity for products of indepedent expectation values <a href="https://hef.ru.nl/~tbudd/mct/lectures/probability_random_variables.html#equation-product-expectation">https://hef.ru.nl/~tbudd/mct/lectures/probability_random_variables.html#equation-product-expectation</a>, and the definition of $\phi_X(t) := \mathbb{E}\left[ e^{itX} \right]$.</p>
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<p>Next, we will use the Taylor expansion around $t = 0$ as is given above, and, for the latter exponential, $\mathbb{E}(X) = 3$ for $\alpha = 3/2, b = 1$ as given.</p>
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\begin{align}
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\phi_{Z_n}(t) &= \left( \phi_X(cn^{-2/3}t) \right)^n \mathbb{E}\left[ e^{itcn^{1/3}\mathbb{E}[X])} \right]
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\\ &= \left( 1 + 3 i cn^{-2/3}t - (|cn^{-2/3}t|+i cn^{-2/3}t)\,\sqrt{2\pi|cn^{-2/3}t|} + \mathcal{O}(t^2) \right)^n e^{3itcn^{1/3}}
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\\ &= \left( 1 + \frac{1}{n} \left[ 3 i cn^{1/3}t - (|ct|+i ct)\,\sqrt{2\pi|ct|} \right] + \mathcal{O}(t^2) \right)^n e^{3itcn^{1/3}}
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\end{align}<p>Taking the limit $n \to \infty$, the first set of parentheses can be rewritten in terms of an exponential using the identity $\lim_{n\to\infty} (1 + \frac{a}{n})^n = e^{a}$, we find a way to our desired expression.</p>
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\begin{align}
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\lim_{n\to\infty} \phi_{Z_n}(t) &= \lim_{n\to\infty} \left( 1 + \frac{1}{n} \left[ 3 i cn^{1/3}t - (|ct|+i ct)\,\sqrt{2\pi|ct|} \right] + \mathcal{O}(t^2) \right)^n e^{-3itcn^{1/3}}
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\\ &= \lim_{n\to\infty} \exp{({3 i cn^{1/3}t - (|ct|+i ct)\,\sqrt{2\pi|ct|}})} \exp{(e^{-3itcn^{1/3}})}
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\\ &= \exp{({-(|ct|+i ct)\,\sqrt{2\pi|ct|}})}
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\end{align}<p>This matches $\phi_S(t) = \exp\big(-(|t|+it)\sqrt{|t|}\big)$ for $\sqrt{2\pi} c^{3/2} = 1 \implies c = (2\pi)^{\frac{-1}{3}}$.</p>
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<p><strong>(c)</strong> The random variable $\mathcal{S}$ has a <a href="https://en.wikipedia.org/wiki/Stable_distribution">stable Lévy distribution</a> with index $\alpha = 3/2$ and skewness $\beta = 1$. Its probability density function $f_{\mathcal{S}}(x)$ does not admit a simple expression, but can be accessed numerically using SciPy's <code>scipy.stats.levy_stable.pdf(x,1.5,1.0)</code>. Verify numerically that the generalized CLT of part (b) holds by comparing an appropriate histogram to this PDF. <strong>(10 pts)</strong></p>
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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-e08d054985cfa762"></a><a name="cell-e08d054985cfa762"></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="kn">from</span> <span class="nn">scipy.stats</span> <span class="kn">import</span> <span class="n">levy_stable</span>
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<span class="k">def</span> <span class="nf">sample_Zn</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">n</span><span class="p">):</span>
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<span class="k">assert</span> <span class="n">n</span> <span class="o">>=</span> <span class="mi">1</span> <span class="ow">and</span> <span class="nb">type</span><span class="p">(</span><span class="n">n</span><span class="p">)</span> <span class="o">==</span> <span class="nb">int</span>
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<span class="n">E_X</span> <span class="o">=</span> <span class="n">alpha</span><span class="o">*</span><span class="n">b</span><span class="o">/</span><span class="p">(</span><span class="n">alpha</span> <span class="o">-</span> <span class="mi">1</span><span class="p">)</span>
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<span class="n">samples</span> <span class="o">=</span> <span class="p">[</span><span class="n">inversion_sample</span><span class="p">(</span><span class="k">lambda</span> <span class="n">p</span><span class="p">:</span> <span class="n">f_inv_pareto</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">,</span> <span class="n">p</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">c</span><span class="o">*</span><span class="n">n</span><span class="o">**</span><span class="p">(</span><span class="mi">1</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span><span class="o">*</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">samples</span><span class="p">)</span> <span class="o">-</span> <span class="n">E_X</span><span class="p">)</span>
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<span class="n">alpha</span> <span class="o">=</span> <span class="mi">3</span><span class="o">/</span><span class="mi">2</span>
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<span class="n">beta</span> <span class="o">=</span> <span class="mi">1</span>
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<span class="n">c</span> <span class="o">=</span> <span class="p">(</span><span class="mi">2</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="p">(</span><span class="o">-</span><span class="mi">1</span><span class="o">/</span><span class="mi">3</span><span class="p">)</span>
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<span class="n">n</span> <span class="o">=</span> <span class="mi">1000</span>
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<span class="n">pdf</span> <span class="o">=</span> <span class="k">lambda</span> <span class="n">x</span><span class="p">:</span> <span class="n">levy_stable</span><span class="o">.</span><span class="n">pdf</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">alpha</span><span class="p">,</span> <span class="n">beta</span><span class="p">)</span>
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<span class="n">samples</span> <span class="o">=</span> <span class="p">[</span><span class="n">sample_Zn</span><span class="p">(</span><span class="n">alpha</span><span class="p">,</span> <span class="n">b</span><span class="p">,</span> <span class="n">c</span><span class="p">,</span> <span class="n">n</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>
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<span class="n">compare_plot</span><span class="p">(</span><span class="n">samples</span><span class="p">,</span> <span class="n">pdf</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
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</pre></div>
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<b>Comments:</b> <p>You are a bit overfitting.</p>
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<img 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<h2 id="Joint-probability-density-functions-and-sampling-the-normal-distribution">Joint probability density functions and sampling the normal distribution<a class="anchor-link" href="#Joint-probability-density-functions-and-sampling-the-normal-distribution">¶</a></h2><p><strong>(30 Points)</strong></p>
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<p>Let $\Phi$ be a uniform random variable on $(0,2\pi)$ and $R$ an independent continuous random variable with probability density function $f_R(r) = r\,e^{-r^2/2}$ for $r>0$. Set $X = R \cos \Phi$ and $Y = R \sin \Phi$. This is called the <strong>Box-Muller transform</strong>.</p>
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<p><strong>(a)</strong> Since $\Phi$ and $R$ are independent, the joint probability density of $\Phi$ and $R$ is $f_{\Phi,R}(\phi,r) = f_\Phi(\phi)f_R(r) = \frac{1}{2\pi}\, r\,e^{-r^2/2}$. Show by change of variables that $X$ and $Y$ are also independent and both distributed as a standard normal distribution $\mathcal{N}$. <strong>(15 pts)</strong></p>
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<a name="comment-cell-4f20e3b730ba0d23"></a><a name="cell-4f20e3b730ba0d23"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><span class="pull-right">
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Score: 15.0 / 15.0 <a href="#top">(Top)</a>
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<p>The coordinate transformation $T$ is defined as $T(\Phi, R) = (R\cos{\Phi}, R\sin{\Phi}) = (X, Y)$. As $T$ is invertible differentiable, we can write the equality between the joint probability density in both coordinate pairs as follows, using the Jacobian.
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f_{X,Y}(x,y) \Big|\frac{\mathrm{d}x}{\mathrm{d}\phi}\frac{\mathrm{d}y}{\mathrm{d}r}-\frac{\mathrm{d}y}{\mathrm{d}\phi}\frac{\mathrm{d}x}{\mathrm{d}r}\Big|
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= f_{X,Y}(T(\phi,r)) \Big|\frac{\mathrm{d}x}{\mathrm{d}\phi}\frac{\mathrm{d}y}{\mathrm{d}r}-\frac{\mathrm{d}y}{\mathrm{d}\phi}\frac{\mathrm{d}x}{\mathrm{d}r}\Big|
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= f_{X,Y}(T(\phi,r)) \Big|-r\sin{\phi}\sin{\phi}-r\cos{\phi}\cos{\phi}\Big|
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= f_{X,Y}(T(\phi,r)) r
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= f_{\Phi,R}(\phi,r)
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= \frac{1}{2\pi}\, r\,e^{-r^2/2}
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\\ \implies f_{X,Y}(x,y) = \frac{1}{2\pi}\,e^{-r^2/2} = \frac{1}{\sqrt{2\pi}}e^{-x^2/2}\frac{1}{\sqrt{2\pi}}e^{-y^2/2} = f_{X}(x)f_Y(y)
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using that $r^2 = x^2 + y^2$ in the second to last step. We conclude that $X$ and $Y$ are independent, and the factorization shows that they are both distributed as a standard normal distribution $\mathcal{N}(0,1)$.</p>
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<p><strong>(b)</strong> Write a function to sample a pair of independent normal random variables using the Box-Muller transform. Hint: to sample $R$ you can use the inversion method of the first exercise. Produce a histogram to check the distribution of your normal variables. <strong>(15 pts)</strong></p>
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<p>For the sampling of $R$, we take its PDF, calculate its CDF, invert it, and use the function <code>inversion_sample</code> to pull values for $R$.</p>
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$$
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f_R(r) = re^{-r^2/2}
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\\ \implies F_R(r) = \int_{0}^r te^{-t^2/2}dt = 1-e^{-r^2/2}
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$$<p>integrating over values $r > 0$ as it cannot be negative, and using a substitution with $z := t^2$.</p>
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<p>Now the inversion.</p>
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$$
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p := F_R(r) = 1-e^{-r^2/2} \implies r = \sqrt{-2\ln{(1-p)}}
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$$<p>for $p \in [0, 1]$. Do note that we can also calculate $r = \sqrt{-2\ln{(p)}}$ as $p$ and $1 - p$ are identically distributed, also keeping the argument to $\ln$ positive, saving just one calculation, although I do not use this in the following.</p>
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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-9bf8873cce1d179c"></a><a name="cell-9bf8873cce1d179c"></a><div class="panel-heading"><span class="nbgrader-label">Student's answer</span><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">def</span> <span class="nf">random_normal_pair</span><span class="p">():</span>
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<span class="sd">'''Return two independent normal random variables.'''</span>
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<span class="n">phi</span> <span class="o">=</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="mi">2</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="n">r</span> <span class="o">=</span> <span class="n">inversion_sample</span><span class="p">(</span><span class="k">lambda</span> <span class="n">p</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="o">-</span><span class="mi">2</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="mi">1</span><span class="o">-</span><span class="n">p</span><span class="p">)))</span>
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<span class="n">x</span><span class="p">,</span> <span class="n">y</span> <span class="o">=</span> <span class="n">r</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">phi</span><span class="p">),</span> <span class="n">r</span><span class="o">*</span><span class="n">np</span><span class="o">.</span><span class="n">sin</span><span class="p">(</span><span class="n">phi</span><span class="p">)</span>
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<span class="k">return</span> <span class="n">x</span><span class="p">,</span> <span class="n">y</span>
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<span class="c1"># Plotting</span>
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<span class="n">pdf</span> <span class="o">=</span> <span class="n">gaussian</span>
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<span class="n">samples</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="n">random_normal_pair</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">100000</span><span class="p">)])</span>
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<span class="k">for</span> <span class="n">index</span> <span class="ow">in</span> <span class="p">[</span><span class="mi">0</span><span class="p">,</span> <span class="mi">1</span><span class="p">]:</span>
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<span class="n">compare_plot</span><span class="p">(</span><span class="n">samples</span><span class="p">[:,</span><span class="n">index</span><span class="p">],</span> <span class="n">pdf</span><span class="p">,</span> <span class="o">-</span><span class="mi">5</span><span class="p">,</span> <span class="mi">5</span><span class="p">,</span> <span class="mi">100</span><span class="p">)</span>
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