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alias: probability mass function

Real random variable

Discrete random variable

A discrete random variable is a Real random variable with a range 𝑅 of cardinality ≤ℵ0, prob that is a set of either finite size or with a bijection to the natural numbers. For any discrete random variable 𝑋 it is possible to define a probability mass function 𝑝𝑋 :ℝ →ℝ such that

𝑝𝑋(𝑥)=ℙ(𝑋=𝑥)

from which it follows the series

∑𝑥∈supp(𝑋)𝑝𝑋(𝑥)=1

Similarly we may define the Cumulative distribution function of a discrete random variable (See Variable distribution equivalence).

Distributions

The most common distributions of discrete random variables are

• Bernoulli trial
• Binomial distribution
• Hypergeometric distribution
• Discrete uniform distribution
• Geometric distribution
• Negative binomial distribution
• Negative hypergeometric distribution
• Poisson distribution

	with replacement	without replacement
fixed trials	Binomial distribution	Hypergeometric distribution
fixed successes	Negative binomial distribution	Negative hypergeometric distribution
until success	Geometric distribution

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