# Discrete distribution

2010 Mathematics Subject Classification: Primary: 60E05 [MSN][ZBL]

A probability distribution (concentrated) on a finite or countably infinite set of points of a sampling space $\Omega$. More exactly, let $\omega _ {1} , \omega _ {2} \dots$ be the sample points and let

$$\tag{1 } p _ {i} = p ( \omega _ {i} ) ,\ \ i= 1 , 2 \dots$$

be numbers satisfying the conditions

$$\tag{2 } p _ {i} \geq 0 ,\ \sum _ { i } p _ {i} = 1 .$$

Relations (1) and (2) fully define a discrete distribution on the space $\Omega$, since the probability measure of any set $A \subset \Omega$ is defined by the equation

$$P ( A) = \sum _ {\{ {i } : {\omega _ {i} \in A } \} } p _ {i} .$$

Accordingly, the distribution of a random variable $X ( \omega )$ is said to be discrete if it assumes, with probability one, a finite or a countably infinite number of distinct values $x _ {i}$ with probabilities $p _ {i} = {\mathsf P} \{ \omega : {X ( \omega ) = x _ {i} } \}$. In the case of a distribution on the real line, the distribution function $F ( x) = \sum _ {\{ {i } : {x _ {i} < x } \} } p _ {i}$ has jumps at the points $x _ {i}$ equal to $p _ {i} = F ( x _ {i} + 0 ) - F ( x _ {i} )$, and is constant in the intervals $[ x _ {i} , x _ {i+} 1 )$. The following discrete distributions occur most frequently: the binomial distribution, the geometric distribution, the hypergeometric distribution, the negative binomial distribution, the multinomial distribution, and the Poisson distribution.

A word of caution. In the Russian literature, $F ( x) = {\mathsf P} \{ X < x \}$, whereas in Western literature $F ( x) = {\mathsf P} \{ X \leq x \}$. So the distribution functions are slightly different: left continuous in the Russian literature, and right continuous in the Western literature.