# DISCRETE DISTRIBUTIONS

Discrete distributions are used to describe random variables that can only take countably many different values. In this chapter, we present several discrete probability distributions and their properties.

# 18.1 Basic Concepts and Facts

Definition 18.1 (Bernoulli Distribution). Let Ω = {0, 1} and p (0, 1). A random variable X on (Ω, 2Ω) is said to have a Bernoulli distribution with parameter p, written as X ~ Be(p), if and only if

Definition 18.2 (Binomial Distribution). Let Ω = {0, 1, 2,…, n}, where n is a positive integer, and p (0, 1). A random variable X on (Ω, 2Ω) is said to have a binomial distribution with parameters (n, p), written as X ~ B(n, p), if and only if

where q = 1 − p and

When n = 1, the binomial distribution is the same as the Bernoulli distribution.

Definition 18.3 (Poisson Distribution). Let Ω = {0, 1, 2,…} and θ > 0. A random variable X on (Ω, 2Ω) is said to have a Poisson distribution with parameter θ, written as X ~ P(θ), if and ...

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