Definition:Discrete Uniform Distribution

Definition

Let $X$ be a discrete random variable on a probability space.

Then $X$ has a discrete uniform distribution with parameter $n$ if:

• $\operatorname{Im} \left({X}\right) = \left\{{1, 2, \ldots, n}\right\}$

• $\Pr \left({X = k}\right) = \dfrac 1 n$

That is, there is a number of outcomes, and

This is written:

$X \sim \operatorname{U} \left({n}\right)$

This distribution trivially gives rise to a probability mass function satisfying $\Pr \left({\Omega}\right) = 1$, because:

$\displaystyle \sum_{k \in \Omega_X} \frac 1 n = \sum_{k = 1}^n \frac 1 n = n \frac 1 n = 1$

Thus it serves as a model for a discrete probability space with equiprobable outcomes.