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<h1>
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Day 4 - Binomial and geometric distributions
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</h1>
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<time class="published" datetime="2018-11-11T10:19:00+01:00">
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11 novembre 2018
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</time>
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</li>
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<li>Blog</li>
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<li>Basics</li>
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<h2>Binomial distribution</h2>
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<h3>Problem 1</h3>
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<p>The ratio of boys to girls for babies born in Russia is <span class="math">\(r=\frac{N_b}{N_g}=1.09\)</span>. If there is 1 child born per birth, what proportion of Russian families with exactly 6 children will have at least 3 boys?</p>
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<h3>Mathematical explanation</h3>
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<p>Let's first compute the probability of having a boy :</p>
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<div class="math">$$p_b=\frac{N_b}{N_b+N_g}$$</div>
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<p>where:
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* <span class="math">\(N_b\)</span> is the number of boys
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* <span class="math">\(N_g\)</span> is the number of girls
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* <span class="math">\(r=\frac{N_b}{N_g}\)</span></p>
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<div class="math">$$p_b=\frac{1}{1+\frac{1}{r}}$$</div>
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<div class="math">$$p_b=\frac{r}{r+1}$$</div>
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<div class="highlight"><pre><span></span><span class="n">r</span> <span class="o">=</span> <span class="mf">1.09</span>
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<span class="n">p_b</span><span class="o">=</span><span class="n">r</span><span class="o">/</span><span class="p">(</span><span class="n">r</span><span class="o">+</span><span class="mi">1</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"The probability of having a boy is p=</span><span class="si">{p_b:3f}</span><span class="s2">"</span><span class="p">)</span>
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</pre></div>
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<div class="highlight"><pre><span></span><span class="err">The probability of having a boy is p=0.521531</span>
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</pre></div>
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<p>The probability of getting 3 boys in 6 children is given by :
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</p>
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<div class="math">$$b(x=3, n=6, p=p_b)$$</div>
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<p>In order to compute the proportion of Russian families with exactly 6 children will have at 3 least boys we need to compute the cumulative probability distribution </p>
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<div class="math">$$b(x\geq 3, n=6, p=p_b) = \sum_{i=3}^{6} b(x\geq i, n=6, p=p_b)$$</div>
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<h3>Let's code it !</h3>
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<div class="highlight"><pre><span></span><span class="kn">import</span> <span class="nn">math</span>
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<span class="k">def</span> <span class="nf">bi_dist</span><span class="p">(</span><span class="n">x</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">):</span>
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<span class="n">b</span> <span class="o">=</span> <span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">math</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">x</span><span class="p">)</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">factorial</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="n">x</span><span class="p">)))</span><span class="o">*</span><span class="p">(</span><span class="n">p</span><span class="o">**</span><span class="n">x</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">p</span><span class="p">)</span><span class="o">**</span><span class="p">(</span><span class="n">n</span><span class="o">-</span><span class="n">x</span><span class="p">))</span>
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<span class="k">return</span><span class="p">(</span><span class="n">b</span><span class="p">)</span>
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<span class="n">b</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="n">p_b</span><span class="p">,</span> <span class="mi">6</span>
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<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="mi">3</span><span class="p">,</span><span class="mi">7</span><span class="p">):</span>
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<span class="n">b</span> <span class="o">+=</span> <span class="n">bi_dist</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"probability of getting at least 3 boys in a family with exactly 6 children : </span><span class="si">{b:.3f}</span><span class="s2">"</span><span class="p">)</span>
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</pre></div>
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<div class="highlight"><pre><span></span><span class="err">probability of getting at least 3 boys in a family with exactly 6 children : 0.696</span>
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</pre></div>
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<h3>Problem 2</h3>
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<p>A manufacturer of metal pistons finds that, 12% on average, of the pistons they manufacture are rejected because they are incorrectly sized. What is the probability that a batch of 10 pistons will contain:
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* No more than 2 rejects?
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* At least 2 rejects?</p>
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<h3>Mathematical explanation</h3>
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<p>On average 12% of the pistons are rejected, this means that a piston has a probability of <span class="math">\(p_{rejected}=0.12\)</span> to be rejected.</p>
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<p>The probability of getting less than 2 faulty pistons in a batch is :
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</p>
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<div class="math">$$p(rejet<2) = b(x\leq 2, n= 10, p=p_{rejected})$$</div>
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<div class="math">$$p(rejet<2) = \sum_{i=0}^{2} b(x\leq i, n=10, p=p_{rejected})$$</div>
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<div class="highlight"><pre><span></span><span class="n">b</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">12</span><span class="o">/</span><span class="mi">100</span><span class="p">,</span> <span class="mi">10</span>
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<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="mi">0</span><span class="p">,</span><span class="mi">3</span><span class="p">):</span>
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<span class="n">b</span> <span class="o">+=</span> <span class="n">bi_dist</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"The probability of getting less than 2 faulty pistons in a batch is : </span><span class="si">{b:.3f}</span><span class="s2">"</span><span class="p">)</span>
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</pre></div>
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<div class="highlight"><pre><span></span><span class="err">The probability of getting less than 2 faulty pistons in a batch is : 0.891</span>
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</pre></div>
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<p>The probability that a batch of 10 pistons will contain at least 2 rejects :
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</p>
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<div class="math">$$p(rejet<2) = b(x\geq 2, n= 10, p=p_{rejected})$$</div>
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<div class="math">$$p(rejet<2) = \sum_{i=2}^{10} b(x\geq i, n=10, p=p_{rejected})$$</div>
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<div class="highlight"><pre><span></span><span class="n">b</span><span class="p">,</span> <span class="n">p</span><span class="p">,</span> <span class="n">n</span> <span class="o">=</span> <span class="mi">0</span><span class="p">,</span> <span class="mi">12</span><span class="o">/</span><span class="mi">100</span><span class="p">,</span> <span class="mi">10</span>
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<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="mi">2</span><span class="p">,</span><span class="mi">11</span><span class="p">):</span>
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<span class="n">b</span> <span class="o">+=</span> <span class="n">bi_dist</span><span class="p">(</span><span class="n">i</span><span class="p">,</span> <span class="n">n</span><span class="p">,</span> <span class="n">p</span><span class="p">)</span>
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<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"The probability of getting at least 2 faulty pistons in a batch is : </span><span class="si">{b:.3f}</span><span class="s2">"</span><span class="p">)</span>
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</pre></div>
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<div class="highlight"><pre><span></span><span class="err">The probability of getting at least 2 faulty pistons in a batch is : 0.342</span>
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</pre></div>
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<h2>Geometric distribution</h2>
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<h3>Problem 1</h3>
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<p>The probability that a machine produces a defective product is <span class="math">\(\frac{1}{3}\)</span>. What is the probability that the first defect is found during the fith inspection?</p>
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<h3>Mathematical explanation</h3>
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<p>In this case, we will use a geometric distribution to evaluate the probability :
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* <span class="math">\(n=5\)</span>
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* <span class="math">\(p=\frac{1}{3}\)</span></p>
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<p>Hence, the probability that the first defect is found during the fith inspection is <span class="math">\(g(n=5,p=1/3)\)</span></p>
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<div class="highlight"><pre><span></span><span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"The probability that the first defect is found during the fith inspection is {round(((1-p)**(n-1)) * p, 3)}"</span><span class="p">)</span>
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</pre></div>
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<div class="highlight"><pre><span></span><span class="err">The probability that the first defect is found during the fith inspection is 0.038</span>
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</pre></div>
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<h3>Problem 2</h3>
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<p>The probability that a machine produces a defective product is <span class="math">\(\frac{1}{3}\)</span>. What is the probability that the first defect is found during the first 5 inspections?</p>
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<h3>Mathematical explanation</h3>
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<p>In this problem, we need to compute the cumulative distribution function
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</p>
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<div class="math">$$p(x \leq5) = \sum_{i=1}^{5} g(n=i,p=1/3)$$</div>
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<div class="highlight"><pre><span></span><span class="n">p_x5</span> <span class="o">=</span> <span class="mi">0</span>
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<span class="n">p</span><span class="o">=</span><span class="mi">1</span><span class="o">/</span><span class="mi">3</span>
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<span class="n">n</span><span class="o">=</span><span class="mi">5</span>
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<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="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">p_x5</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="n">i</span><span class="o">-</span><span class="mi">1</span><span class="p">)</span> <span class="o">*</span> <span class="n">p</span>
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<span class="nb">print</span><span class="p">(</span><span class="sa">f</span><span class="s2">"The probability that the first defect is found during the first 5 inspection is {round(p_x5, 3)}"</span><span class="p">)</span>
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</pre></div>
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<div class="highlight"><pre><span></span><span class="err">The probability that the first defect is found during the first 5 inspection is 0.868</span>
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</pre></div>
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"MathJax.Hub.Register.StartupHook('SVG Jax Ready',function () {" +
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"var VARIANT = MathJax.OutputJax.SVG.FONTDATA.VARIANT;" +
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"VARIANT['normal'].fonts.unshift('MathJax_default');" +
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"VARIANT['bold'].fonts.unshift('MathJax_default-bold');" +
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"VARIANT['italic'].fonts.unshift('MathJax_default-italic');" +
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"VARIANT['-tex-mathit'].fonts.unshift('MathJax_default-italic');" +
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"});" +
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"}";
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(document.body || document.getElementsByTagName('head')[0]).appendChild(mathjaxscript);
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}</script>
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</div>
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<aside>
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<div class="bug-reporting__panel">
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<h3>Find an error or bug? Have a suggestion?</h3>
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<p>Everything on this site is avaliable on GitHub. Head on over and <a href='https://github.com/redoules/redoules.github.io/issues/new'>submit an issue.</a> You can also message me directly by <a href='mailto:guillaume.redoules@gadz.org'>email</a>.</p>
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</div>
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</aside>
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</section>
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</div>
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<!-- start of footer section -->
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<footer class="footer">
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<div class="container">
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<p class="text-muted">
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<center>This project contains 115 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
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<br/>
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Copyright © Guillaume Redoulès,
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|
<time datetime="2018">2018</time>.
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</center>
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</p>
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</script>
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