Editing Probability distribution (section)

==Random number generation==
{{Main|Pseudo-random number sampling}}

Most algorithms are based on a [[pseudorandom number generator]] that produces numbers <math>X</math> that are uniformly distributed in the [[half-open interval]] {{closed-open|0, 1}}. These [[random variate]]s <math>X</math> are then transformed via some algorithm to create a new random variate having the required probability distribution. With this source of uniform pseudo-randomness, realizations of any random variable can be generated.<ref name=":0">{{Citation|last1=Dekking|first1=Frederik Michel| title=Why probability and statistics?|date=2005|work=A Modern Introduction to Probability and Statistics| pages=1–11| publisher =Springer London|isbn=978-1-85233-896-1|last2=Kraaikamp|first2=Cornelis| last3=Lopuhaä|first3=Hendrik Paul| last4=Meester| first4=Ludolf Erwin| doi=10.1007/1-84628-168-7_1}}</ref>

For example, suppose {{mvar|U}} has a uniform distribution between 0 and 1. To construct a random Bernoulli variable for some {{math|0 < p < 1}}, define
<math display="block">X = \begin{cases}
1& \text{if } U<p\\
0& \text{if } U\geq p.
\end{cases}</math>
We thus have
<math display="block">P(X=1) = P(U<p) = p, \quad P(X=0) = P(U\geq p) = 1-p.</math>
Therefore, the random variable {{mvar|X}} has a Bernoulli distribution with parameter {{mvar|p}}.<ref name=":0"/>

This method can be adapted to generate real-valued random variables with any distribution: for be any cumulative distribution function {{mvar|F}}, let {{math|''F''{{sup|inv}}}} be the generalized left inverse of <math>F,</math> also known in this context as the ''[[quantile function]]'' or ''inverse distribution function'':
<math display="block">F^{\mathrm{inv}}(p) = \inf \{x \in \mathbb{R} : p \le F(x)\}.</math>
Then, {{math|''F''{{sup|inv}}(''p'') ≤ ''x''}} if and only if {{math|''p'' ≤ ''F''(''x'')}}. As a result, if {{mvar|U}} is uniformly distributed on {{math|[0, 1]}}, then the cumulative distribution function of {{math|''X'' {{=}} ''F''{{sup|inv}}(''U'')}} is {{mvar|F}}.

For example, suppose we want to generate a random variable having an exponential distribution with parameter <math>\lambda</math> — that is, with cumulative distribution function <math>F : x \mapsto 1 - e^{-\lambda x}.</math>
<math display="block">\begin{align}
F(x) = u &\Leftrightarrow 1-e^{-\lambda x} = u \\[2pt]
&\Leftrightarrow e^{-\lambda x } = 1-u \\[2pt]
&\Leftrightarrow -\lambda x = \ln(1-u) \\[2pt]
&\Leftrightarrow x = \frac{-1}{\lambda}\ln(1-u)
\end{align}</math>
so <math>F^{\mathrm{inv}}(u) = -\tfrac{1}{\lambda}\ln(1-u)</math>, and if {{mvar|U}} has a uniform distribution on {{math|[0, 1)}} then <math>X = -\tfrac{1}{\lambda}\ln(1-U)</math> has an exponential distribution with parameter <math>\lambda.</math><ref name=":0" />

Although from a theoretical point of view this method always works, in practice the inverse distribution function is unknown and/or cannot be computed efficiently. In this case, other methods (such as the [[Monte Carlo method]]) are used.