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<h1>
Geometric distribution
</h1>
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<time class="published" datetime="2018-11-11T21:19:00+01:00">
11 novembre 2018
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<li>Mathematics</li>
<li>Statistics</li>
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<h2>Negative Binomial Experiment</h2>
<p>A negative binomial experiment is a statistical experiment that has the following properties:</p>
<ul>
<li>The experiment consists of n repeated trials.</li>
<li>The trials are independent.</li>
<li>The outcome of each trial is either success (s) or failure (f).</li>
<li><span class="math">\(P(s)\)</span> is the same for every trial.</li>
<li>The experiment continues until x successes are observed</li>
</ul>
<p>If <span class="math">\(X\)</span> is the number of experiments until the <span class="math">\(x^{th}\)</span> success occures, then <span class="math">\(X\)</span> is a discrete random variable called a negative binomial</p>
<h2>Negative Binomial Distribution</h2>
<p>Consider the following probability mass function:
</p>
<div class="math">$$b^*(x,n,p) = {\binom{n-1}{x-1}}p^xq^{n-x}$$</div>
<p>
The function above is negative binomial and has the following properties:</p>
<ul>
<li>The number of successes to be observed is <span class="math">\(x\)</span>.</li>
<li>The total number of trials is <span class="math">\(n\)</span>.</li>
<li>The probability of success of 1 trial is <span class="math">\(p\)</span>.</li>
<li>The probability of failure of 1 trial <span class="math">\(q\)</span>, where <span class="math">\(q=1-p\)</span>.</li>
<li><span class="math">\(b^*(x,n,p)\)</span> is the <em>negative binomial probability</em>, meaning the probability of having exactly <span class="math">\(x-1\)</span> successes out of <span class="math">\(n-1\)</span> trials and having <span class="math">\(x\)</span> successes after <span class="math">\(n\)</span> trials.</li>
</ul>
<h2>Geometric Distribution</h2>
<p>The geometric distribution is a special case of the negative binomial distribution that deals with the number of Bernoulli trials required to get a success (i.e., counting the number of failures before the first success). Recall that <span class="math">\(X\)</span> is the number of successes in <span class="math">\(n\)</span> independent Bernoulli trials, so for each <span class="math">\(i\)</span> (where $1\leq i\leq n):</p>
<p>$ X_i = </p>
<div class="math">\begin{cases}
1 if the i^{th} trial is a success \\
0 otherwise x
\end{cases}</div>
<p>
$</p>
<p>The geometric distribution is a negative binomial distribution where the number of successes is 1. We express this with the following formula:</p>
<div class="math">$$g(n,p)=q^{n-1}p$$</div>
<h3>Example</h3>
<p>Bob is a high school basketball player. He is a 70% free throw shooter, meaning his probability of making a free throw is 0.7. What is the probability that Bob makes his first free throw on his fifth shot?</p>
<p>For this experiment n=5, p=0.7 and q=0.3
So :
</p>
<div class="math">$$g(n=5, p=0.7)=0.3^4 0.7=0.00567$$</div>
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