Uniform Distribution

Similar to the Discrete Uniform Distribution, but with a continuous range for the random variable $X$

Probability Density Function

A random variable has a uniform distribution $X \sim u n i f (α, β)$ if and only if it has the pdf

f (x; α, β) = \frac{1}{β - α}, α \leq x \leq β

Mean

\begin{aligned} μ = E (X) & = \int_{α}^{β} x \frac{1}{β - α} d x \\ = \frac{1}{β - α} {[\frac{x^{2}}{2} |}_{α}^{β} \\ = \frac{α + β}{2} \end{aligned}

Variance

\begin{aligned} E (X^{2}) & = \int_{α}^{β} x^{2} \frac{1}{β - α} \\ \dots \\ = \frac{β^{2} + α β + α^{2}}{3} \\ V a r (X) & = E (X^{2}) - [E (X)]^{2} \\ \dots \\ = \frac{1}{12} (α - β)^{2} \end{aligned}

CDF

F (x) \int_{α}^{x} \frac{1}{β - α} = \frac{x - α}{β - α}

As possible values of x depend on $α, β$ this distribution is called a irregular case