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Uniform Distribution

NIST/SEMATECH Section 1.3.6.6.2 Uniform Distribution

PDF

Uniform(-1, 1) PDF -1 -0.5 0 0.5 1 x 0 0.1 0.2 0.3 0.4 0.5 f(x)

CDF

Uniform(-1, 1) CDF -1 -0.5 0 0.5 1 x 0 0.2 0.4 0.6 0.8 1 F(x)

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CDF

Formulas

Probability Density Function

f(x)={1b−aa≤x≤b0otherwisef(x) = \begin{cases} \frac{1}{b - a} & a \le x \le b \\ 0 & \text{otherwise} \end{cases}

Cumulative Distribution Function

F(x)={0x<ax−ab−aa≤x≤b1x>bF(x) = \begin{cases} 0 & x < a \\ \frac{x - a}{b - a} & a \le x \le b \\ 1 & x > b \end{cases}

Properties

Mean

a+b2\frac{a + b}{2}

Variance

(b−a)212\frac{(b - a)^2}{12}

Overview

The uniform distribution assigns equal probability to all values within a specified interval [a,b][a, b]. It is the simplest continuous distribution and serves as a baseline for random number generation.

Related Distributions

Beta Distribution Normal Distribution