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<h1>
Day 6 - The Central Limit Theorem
</h1>
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<time class="published" datetime="2018-11-13T22:15:00+01:00">
13 novembre 2018
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<h2>Problem 1</h2>
<p>A large elevator can transport a maximum of <span class="math">\(9800\)</span> kg. Suppose a load of cargo containing <span class="math">\(49\)</span> boxes must be transported via the elevator. The box weight of this type of cargo follows a distribution with a mean of <span class="math">\(\mu=205\)</span> kg and a standard deviation of <span class="math">\(\sigma=15\)</span> kg. Based on this information, what is the probability that all boxes can be safely loaded into the freight elevator and transported?</p>
<h3>Mathematical explanation</h3>
<p>This problem can be tackled with the central limit theorem. Since the number of boxes is "large", the sum of the weight approaches normal distribution with :
<em> <span class="math">\(\mu' = n\mu\)</span>
</em> <span class="math">\(\sigma'=\sigma\sqrt{n}\)</span></p>
<p>If we want to know the probability of the sum of the mass of all boxes to be under a certain weight, we can compute the cumulative density function :
</p>
<div class="math">$$P(x&lt;9800) = F_X(9800)$$</div>
<div class="highlight"><pre><span></span><span class="n">max_load</span> <span class="o">=</span> <span class="mi">9800</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">49</span>
<span class="n">mu</span> <span class="o">=</span> <span class="mi">205</span>
<span class="n">st_dev</span> <span class="o">=</span> <span class="mi">15</span>
<span class="kn">import</span> <span class="nn">math</span>
<span class="k">def</span> <span class="nf">cumulative</span><span class="p">(</span><span class="n">x</span><span class="p">,</span><span class="n">mean</span><span class="p">,</span><span class="n">sd</span><span class="p">):</span>
<span class="k">return</span> <span class="mf">0.5</span><span class="o">*</span><span class="p">(</span><span class="mi">1</span><span class="o">+</span><span class="n">math</span><span class="o">.</span><span class="n">erf</span><span class="p">((</span><span class="n">x</span><span class="o">-</span><span class="n">mean</span><span class="p">)</span><span class="o">/</span><span class="p">(</span><span class="n">sd</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="mi">2</span><span class="p">))))</span>
<span class="n">mu_group</span> <span class="o">=</span> <span class="n">n</span><span class="o">*</span><span class="n">mu</span>
<span class="n">st_dev_group</span> <span class="o">=</span> <span class="n">st_dev</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">f</span><span class="s2">&quot;Probability that all the boxes can be lifted by the elevator : {cumulative(max_load, mu_group, st_dev_group)}&quot;</span><span class="p">)</span>
</pre></div>
<div class="highlight"><pre><span></span>Probability that all the boxes can be lifted by the elevator : 0.009815328628645315
</pre></div>
<h2>Problem 2</h2>
<p>The number of tickets purchased by each student for the University X vs. University Y football game follows a distribution that has a mean of <span class="math">\(\mu=2.4\)</span> and a standard deviation of <span class="math">\(\sigma=2.0\)</span>.</p>
<p>A few hours before the game starts, <span class="math">\(100\)</span> eager students line up to purchase last-minute tickets. If there are only <span class="math">\(250\)</span> tickets left, what is the probability that all students will be able to purchase tickets?</p>
<h3>Mathematical explanation</h3>
<p>We want to know if the sum of all the purchases will exceed the total supply of tickets. Since each student buy follows a Normal distribution and that the number of students is relatively high, the probability that all the students will be able to buy a ticket can be computed by applying the central limit theorem.
The total number of tickets bought follows a normal distribution of mean <span class="math">\(\mu'=n*\mu\)</span> and of standard deviation <span class="math">\(\sigma'=\sigma\sqrt{n}\)</span></p>
<div class="highlight"><pre><span></span><span class="n">ticket_supply</span> <span class="o">=</span> <span class="mi">250</span>
<span class="n">n_students</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">mu</span> <span class="o">=</span> <span class="mf">2.4</span>
<span class="n">st_dev</span> <span class="o">=</span> <span class="mi">2</span>
<span class="n">mu_group</span> <span class="o">=</span> <span class="n">n_students</span><span class="o">*</span><span class="n">mu</span>
<span class="n">st_dev_group</span> <span class="o">=</span> <span class="n">st_dev</span><span class="o">*</span><span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">n_students</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="n">f</span><span class="s2">&quot;Probability that all the students can purchase tickets : {cumulative(ticket_supply, mu_group, st_dev_group)}&quot;</span><span class="p">)</span>
</pre></div>
<div class="highlight"><pre><span></span>Probability that all the students can purchase tickets : 0.691462461274013
</pre></div>
<h2>Problem 3</h2>
<p>You have a sample of <span class="math">\(100\)</span> values from a population with mean <span class="math">\(\mu=500\)</span> and with standard deviation <span class="math">\(\sigma=80\)</span>. Compute the interval that covers the middle <span class="math">\(95%\)</span> of the distribution of the sample mean; in other words, compute <span class="math">\(A\)</span> and <span class="math">\(B\)</span> such that <span class="math">\(P(A&lt;x&lt;B)=0.95\)</span>. Use the value of <span class="math">\(z=1.96\)</span>. Note that is the z-score.</p>
<h3>Mathematical explanation</h3>
<p>The margin of error can be computed with :
</p>
<div class="math">$$MoE = \frac{z-\sigma}{\sqrt{n}}$$</div>
<p>Knowing this, we can figure out what values for x paint the exact middle of the distribution for 0.95 probability, that means theres a 0.0025 leftover on both sides to make the total of 1.</p>
<div class="highlight"><pre><span></span><span class="n">zScore</span> <span class="o">=</span> <span class="mf">1.96</span>
<span class="n">std</span> <span class="o">=</span> <span class="mi">80</span>
<span class="n">n</span> <span class="o">=</span> <span class="mi">100</span>
<span class="n">mean</span> <span class="o">=</span> <span class="mi">500</span>
<span class="n">marginOfError</span> <span class="o">=</span> <span class="n">zScore</span> <span class="o">*</span> <span class="n">std</span> <span class="o">/</span> <span class="n">math</span><span class="o">.</span><span class="n">sqrt</span><span class="p">(</span><span class="n">n</span><span class="p">);</span>
<span class="k">print</span><span class="p">(</span><span class="s2">&quot;A =&quot;</span><span class="p">,</span><span class="n">mean</span><span class="o">-</span> <span class="n">marginOfError</span><span class="p">)</span>
<span class="k">print</span><span class="p">(</span><span class="s2">&quot;B =&quot;</span><span class="p">,</span><span class="n">mean</span> <span class="o">+</span> <span class="n">marginOfError</span><span class="p">)</span>
</pre></div>
<div class="highlight"><pre><span></span>A = 484.32
B = 515.68
</pre></div>
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<li><a href="./blog/Statistics_10days-day5.html">11/18 Day 5 - Poisson and Normal distributions</a></li>
<li><a href="./blog/Statistics_10days-day4.html">11/18 Day 4 - Binomial and geometric distributions</a></li>
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linux/Reloading_.bashrc.html
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linux/free_disk_space_linux.html
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linux/get_ip_linux.html
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<center>This project contains 60 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<br/>
Copyright &copy; Guillaume Redoulès,
<time datetime="2018">2018</time>.

2
python/updating_all_python_package_with_anaconda.html
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@ -151,7 +151,7 @@
<footer class="footer">
<div class="container">
<p class="text-muted">
<center>This project contains 59 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 60 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<br/>
Copyright &copy; Guillaume Redoulès,
<time datetime="2018">2018</time>.

2
search.html
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@ -112,7 +112,7 @@
<footer class="footer">
<div class="container">
<p class="text-muted">
<center>This project contains 59 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 60 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<br/>
Copyright &copy; Guillaume Redoulès,
<time datetime="2018">2018</time>.

9
sitemap.xml
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@ -5,7 +5,7 @@ xmlns="http://www.sitemaps.org/schemas/sitemap/0.9">
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<changefreq>daily</changefreq>
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<changefreq>monthly</changefreq>
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<lastmod>2018-11-12T13:07:00+01:00</lastmod>

2
sql/Sorting_results.html
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@ -344,7 +344,7 @@ Done.
<footer class="footer">
<div class="container">
<p class="text-muted">
<center>This project contains 59 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 60 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<br/>
Copyright &copy; Guillaume Redoulès,
<time datetime="2018">2018</time>.

2
sql/WHERE_SQL_keywords.html
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@ -631,7 +631,7 @@ Done.
<footer class="footer">
<div class="container">
<p class="text-muted">
<center>This project contains 59 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 60 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<br/>
Copyright &copy; Guillaume Redoulès,
<time datetime="2018">2018</time>.

2
sql/display_table.html
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@ -216,7 +216,7 @@ Done.
<footer class="footer">
<div class="container">
<p class="text-muted">
<center>This project contains 59 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 60 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<br/>
Copyright &copy; Guillaume Redoulès,
<time datetime="2018">2018</time>.

2
sql/display_table_filter.html
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@ -346,7 +346,7 @@ Done.
<footer class="footer">
<div class="container">
<p class="text-muted">
<center>This project contains 59 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<center>This project contains 60 pages and is available on <a href="https://github.com/redoules/redoules.github.io">GitHub</a>.
<br/>
Copyright &copy; Guillaume Redoulès,
<time datetime="2018">2018</time>.

2
tipuesearch_content.json
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