Editing Pseudorandom number generator (section)

==Non-uniform generators==
{{Main article|Pseudo-random number sampling}}
Numbers selected from a non-uniform probability distribution can be generated using a [[Uniform distribution (continuous)|uniform distribution]] PRNG and a function that relates the two distributions.

First, one needs the [[cumulative distribution function]] <math>F(b)</math> of the target distribution <math>f(b)</math>:

:<math>F(b)=\int_{-\infty}^b f(b') \, db'</math>

Note that <math>0=F(-\infty)\leq F(b) \leq F(\infty)=1</math>.  Using a random number ''c'' from a uniform distribution as the probability density to "pass by", we get

:<math>F(b)=c</math>

so that

:<math>b=F^{-1}(c)</math>

is a number randomly selected from distribution <math>f(b)</math>. This is based on the [[inverse transform sampling]].

For example, the inverse of cumulative [[Gaussian distribution]] <math>\operatorname{erf}^{-1}(x)</math> with an ideal uniform PRNG with range (0, 1) as input <math>x</math> would produce a sequence of (positive only) values with a Gaussian distribution; however

* When using practical number representations, the infinite "tails" of the distribution have to be truncated to finite values.
* Repetitive recalculation of <math>\operatorname{erf}^{-1}(x)</math> should be reduced by means such as [[ziggurat algorithm]] for faster generation.

Similar considerations apply to generating other non-uniform distributions such as [[Rayleigh distribution|Rayleigh]] and [[Poisson distribution|Poisson]].