Definition:Uniform Distribution/Discrete

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

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

$\Img X = \set {1, 2, \ldots, n}$
$\map \Pr {X = k} = \dfrac 1 n$

That is, there is a number of outcomes with an equal probability of occurrence.

This is written:

$X \sim \DiscreteUniform n$

This distribution trivially gives rise to a probability mass function satisfying $\map \Pr \Omega = 1$, because:

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

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

Also see

  • Results about the discrete uniform distribution can be found here.

Technical Note

The $\LaTeX$ code for \(\DiscreteUniform {n}\) is \DiscreteUniform {n} .

When the argument is a single character, it is usual to omit the braces:

\DiscreteUniform n