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<title>GNU Scientific Library &ndash; Reference Manual: Random Number Distribution Introduction</title>

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<a name="Random-Number-Distribution-Introduction"></a>
<div class="header">
<p>
Next: <a href="The-Gaussian-Distribution.html#The-Gaussian-Distribution" accesskey="n" rel="next">The Gaussian Distribution</a>, Up: <a href="Random-Number-Distributions.html#Random-Number-Distributions" accesskey="u" rel="up">Random Number Distributions</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
</div>
<hr>
<a name="Introduction-3"></a>
<h3 class="section">20.1 Introduction</h3>

<p>Continuous random number distributions are defined by a probability
density function, <em>p(x)</em>, such that the probability of <em>x</em>
occurring in the infinitesimal range <em>x</em> to <em>x+dx</em> is <em>p dx</em>.
</p>
<p>The cumulative distribution function for the lower tail <em>P(x)</em> is
defined by the integral,
</p>
<div class="example">
<pre class="example">P(x) = \int_{-\infty}^{x} dx' p(x')
</pre></div>

<p>and gives the probability of a variate taking a value less than <em>x</em>.
</p>
<p>The cumulative distribution function for the upper tail <em>Q(x)</em> is
defined by the integral,
</p>
<div class="example">
<pre class="example">Q(x) = \int_{x}^{+\infty} dx' p(x')
</pre></div>

<p>and gives the probability of a variate taking a value greater than <em>x</em>.
</p>
<p>The upper and lower cumulative distribution functions are related by
<em>P(x) + Q(x) = 1</em> and satisfy <em>0 &lt;= P(x) &lt;= 1</em>, <em>0 &lt;= Q(x) &lt;= 1</em>.
</p>
<p>The inverse cumulative distributions, <em>x=P^{-1}(P)</em> and <em>x=Q^{-1}(Q)</em> give the values of <em>x</em>
which correspond to a specific value of <em>P</em> or <em>Q</em>.  
They can be used to find confidence limits from probability values.
</p>
<p>For discrete distributions the probability of sampling the integer
value <em>k</em> is given by <em>p(k)</em>, where <em>\sum_k p(k) = 1</em>.
The cumulative distribution for the lower tail <em>P(k)</em> of a
discrete distribution is defined as,
</p>
<div class="example">
<pre class="example">P(k) = \sum_{i &lt;= k} p(i)
</pre></div>

<p>where the sum is over the allowed range of the distribution less than
or equal to <em>k</em>.  
</p>
<p>The cumulative distribution for the upper tail of a discrete
distribution <em>Q(k)</em> is defined as
</p>
<div class="example">
<pre class="example">Q(k) = \sum_{i &gt; k} p(i)
</pre></div>

<p>giving the sum of probabilities for all values greater than <em>k</em>.
These two definitions satisfy the identity <em>P(k)+Q(k)=1</em>.
</p>
<p>If the range of the distribution is 1 to <em>n</em> inclusive then
<em>P(n)=1</em>, <em>Q(n)=0</em> while <em>P(1) = p(1)</em>,
<em>Q(1)=1-p(1)</em>.
</p>
<hr>
<div class="header">
<p>
Next: <a href="The-Gaussian-Distribution.html#The-Gaussian-Distribution" accesskey="n" rel="next">The Gaussian Distribution</a>, Up: <a href="Random-Number-Distributions.html#Random-Number-Distributions" accesskey="u" rel="up">Random Number Distributions</a> &nbsp; [<a href="Function-Index.html#Function-Index" title="Index" rel="index">Index</a>]</p>
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