Math 2.0

definition

Discrete random variable

Definition

Let XX be a real random variable on a probability space (Ω,F,P)(\Omega, \mathcal{F}, P). XX is discrete if there is a countable set S⊆RS \subseteq \mathbb{R} with P(X∈S)=1P(X \in S) = 1. A distribution μ\mu on R\mathbb{R} is discrete if μ(S)=1\mu(S) = 1 for some countable set SS.

Remarks

Then P(X∈B)=∑x∈B∩SP(X=x)P(X \in B) = \sum_{x \in B \cap S} P(X = x) for each Borel set BB. So the probability mass function x↦P(X=x)x \mapsto P(X = x) determines the law of XX. The cumulative distribution function of a discrete random variable is a step function when the set of values has no limit points. It jumps by P(X=x)P(X = x) at each value xx. The Bernoulli, binomial, geometric and Poisson distributions are discrete.

Explanations

L1 human

A discrete random variable takes values that you can list, such as 0,1,2,…0, 1, 2, \dots. The number of heads in ten coin tosses and the number of calls in an hour are discrete random variables.

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Sources

  1. Web https://en.wikipedia.org/wiki/Probability_distribution, Discrete probability distribution

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