# CHAPTER 18

# 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, 2^{Ω}) 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, 2^{Ω}) 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, 2^{Ω}) is said to have a Poisson distribution with parameter θ, written as *X* ~ *P*(θ), if and ...

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