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<title>Random Number Distribution Introduction - GNU Scientific Library -- Reference Manual</title>
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<h3 class="section">20.1 Introduction</h3>
<p>Continuous random number distributions are defined by a probability
density function, p(x), such that the probability of x
occurring in the infinitesimal range x to x+dx is <!-- {$p\,dx$} -->
p dx.
<p>The cumulative distribution function for the lower tail P(x) is
defined by the integral,
and gives the probability of a variate taking a value less than x.
<p>The cumulative distribution function for the upper tail Q(x) is
defined by the integral,
and gives the probability of a variate taking a value greater than x.
<p>The upper and lower cumulative distribution functions are related by
P(x) + Q(x) = 1 and satisfy <!-- {$0 \le P(x) \le 1$} -->
0 <= P(x) <= 1, <!-- {$0 \le Q(x) \le 1$} -->
0 <= Q(x) <= 1.
<p>The inverse cumulative distributions, <!-- {$x=P^{-1}(P)$} -->
x=P^{-1}(P) and <!-- {$x=Q^{-1}(Q)$} -->
x=Q^{-1}(Q) give the values of x
which correspond to a specific value of P or Q.
They can be used to find confidence limits from probability values.
<p>For discrete distributions the probability of sampling the integer
value k is given by p(k), where \sum_k p(k) = 1.
The cumulative distribution for the lower tail P(k) of a
discrete distribution is defined as,
where the sum is over the allowed range of the distribution less than
or equal to k.
<p>The cumulative distribution for the upper tail of a discrete
distribution Q(k) is defined as
giving the sum of probabilities for all values greater than k.
These two definitions satisfy the identity P(k)+Q(k)=1.
<p>If the range of the distribution is 1 to n inclusive then
P(n)=1, Q(n)=0 while P(1) = p(1),
Q(1)=1-p(1).
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