Definition:Uniform Distribution/Discrete

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

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


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


 * $$\Pr \left({X = k}\right) = \frac 1 n$$

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:
 * $$\sum_{k \in \operatorname{Im} \left({X}\right)} \frac 1 n = \sum_{k = 1}^n \frac 1 n = n \frac 1 n = 1$$

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