# CHAPTER 3

# SOME DISCRETE DISTRIBUTIONS

In this chapter we present some frequently used discrete distributions.

## 3.1 DISCRETE UNIFORM, BINOMIAL AND BERNOULLI DISTRIBUTIONS

**Definition 3.1 (Discrete Uniform Distribution)** *A random variable X has a discrete uniform distribution with N points, where N is a positive integer with possible distinct values x _{i}, i* = 1,2, · · · ,

*N, if its probability mass function is given by:*

*If in particular x _{i} = i, i* = 1,2, · · · ,

*N, the probability mass function is shown in Figure 3.1.*

**Theorem 3.1 (Properties of a Discrete Uniform Random Variable)***If X is a random variable having a discrete uniform distribution with N points, then:*

*1.*

*2.*

*3.*

**Corollary 3.1** *If x _{k} = k*,

*k*= 1,2, · · · ,

*N*,

*then:*

*1.*

*2.*

*3.*

*Proof:*

- Left as an exercise for the reader.
- Follows from the definition of the
*mgf. ...*

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