Distribution

The distribution of a variable is a description of the relative numbers of times each possible outcome will occur in a number of trials. The function describing the distribution is called the Probability Function, and the function describing the probability that a given value or any value smaller than it will occur is called the Distribution Function.

Formally, a distribution can be defined as a normalized Measure, and the distribution of a Random Variable $x$ is the Measure $P_x$ on $\Bbb{S}'$ defined by setting

$$P_x(A')=P\{s\in S: x(s)\in A'\},$$

where $(S,\Bbb{S},P)$ is a Probability Space, $(S,\Bbb{S})$ is a Measurable Space, and $P$ a Measure on $\Bbb{S}$ with $P(S)=1$.

See also Continuous Distribution, Discrete Distribution, Distribution Function, Measurable Space, Measure, Probability, Probability Density Function, Random Variable, Statistics

References

Doob, J. L. ``The Development of Rigor in Mathematical Probability (1900-1950).'' Amer. Math. Monthly 103, 586-595, 1996.