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Measure, Probability, and Mathematical Finance: A Problem-Oriented Approach by Hong Xie, Chaoqun Ma, Guojun Gan

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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, 2Ω) is said to have a Bernoulli distribution with parameter p, written as X ~ Be(p), if and only if

equation

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

equation

where q = 1 − p and

equation

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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