Discrete Uniform Distribution

Topics: Probability - Distribution Function

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(definition)

Let $X$ be a discrete random variable such that its range is $R_X=\{x_1,\dots ,x_n\}$ and the occurrence of every one of these elements is equally as probable.

That is, let $X$ such that its distribution function is:

$$f_X(x)=\{\begin{array}{ll}\frac{1}{n}& \text{if }x\in \{x_1,\dots ,x_n\}\\ 0& \text{otherwise}\end{array}$$

In such a case, we say that $X$ has a uniform distribution. We denote this with $X\sim \mathrm{Unif}(\{x_1,\dots ,x_n\})$

Expected Value §

(theorem)

The expected value of such an $X$ is given by:

$$\mathbb{E}[X]=\frac{1}{n}\sum _{x\in R_X}x$$

Variance §

(theorem)

The variance of such an $X$ is given by:

$$\mathrm{Var}(X)=\frac{1}{n}\sum _{x\in R_X}(x-\mathbb{E}[X]{)}^2$$