Discrete distribution

2020 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.

Comments

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.