Uniform Distribution

statistics

Definition

Uniform Distribution

The uniform distribution is a type of probability distribution in which all outcomes are equally likely; each variable has the same probability of occurring. It can be continuous, where any value within a given interval is equally likely, or discrete, where each of several finite outcomes has equal probability.

Probability Density Function

The density $f$ for $X\sim U(a,b)$ ($a,b\in \mathbb{R},a<b$) is given by:

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

Expected Value

The expected value is given by integration:

$$\begin{array}{rl}\mathbb{E}[X]& ={\int }_a^{b}x\cdot f(x)dx\\ & ={\int }_a^b\frac{x}{b-a}dx\\ & =\frac{a+b}{2}\end{array}$$

Variance

The variance is given by integration:

$$\begin{array}{rl}\mathbb{V}ar(X)& =\mathbb{E}[(X-\mathbb{E}[X]{)}^2]\\ & ={\int }_a^b(x-[X]{)}^2\cdot f(x)dx\\ & =\frac{1}{4(b-a)}{\int }_a^b(2x-a-b{)}^{2}dx\\ & =\frac{(b-a{)}^2}{12}\end{array}$$