Characterization of probability distributions

The template {{Expand}} has been deprecated since 26 December 2010, and is retained only for old revisions. If this page is a current revision, please remove the template. In mathematics in general, a characterization theorem says that a particular object – a function, a space, etc. – is the only one that possesses properties specified in the theorem. A characterization of a probability distribution accordingly states that it is the only probability distribution that satisfies specified conditions.

• The assumption that two linear (or non-linear) statistics are identically distributed (or independent, or have a constancy regression and so on) can be used to characterize various populations. For example, according to George Polya's characterization theorem, if $X_{1}$ and $X_{2}$ are independent identically distributed random variables with finite variance, then the statistics

${\begin{aligned}S_{1}&=X_{1}\\[6pt]S_{2}&={\frac {X_{1}+X_{2}}{\sqrt {2}}}\end{aligned}}$

are identically distributed if and only if $X_{1}$ and $X_{2}$ have a normal distribution with zero mean.

• All probability distributions on the half-line [0, ∞) that are memoryless are exponential distributions. "Memoryless" means that if X is a random variable with such a distribution, then for any numbers 0 < y < x,

$\Pr(X>x\mid X>y)=\Pr(X>x-y).\,$

See also

• Characterization (mathematics)