Uniform Distribution

Discrete Uniform Distribution

Definition: Discrete Uniform Distribution

We say that a discrete random variable $X$ has a uniform distribution if its support $S=\{x_1,\dots ,x_n\}$ is finite and its probability mass function $p$ is given by

$$p(x)=\{\begin{array}{l}\frac{1}{n}x\in S\\ 0\text{otherwise}\end{array}$$

NOTATION

$$X\sim U(n)X\in U(n)$$

Continuous Uniform Distribution

Definition: Continuous Uniform Distribution

We say that a continuous random variable has a uniform distribution on the interval $[a;b]$ (or $(a;b)$) if its cumulative distribution function is given by

$$F(x)=\{\begin{array}{l}0\text{ if }x<a\\ \frac{x-a}{b-a}\text{ if }a\le x\le b\\ 1\text{ if }x>b\end{array}$$

NOTATION

$$X\sim U(a,b)X\in U(a,b)$$

Properties

Theorem: Probability Density of Uniform Distributions

The probability density function of a continuous random variable which has the uniform distribution $U(a,b)$ is given by

$$f(x)=\{\begin{array}{l}\frac{1}{b-a}\text{ if }a<x<b\\ 0\text{ otherwise }\end{array}$$

PROOF

TODO