Continuous Uniform Distribution
A continuous random variable $X$ is said to have a Uniform distribution (or rectangular distribution) with parameters $\alpha$ and $\beta$ if its p.d.f. is given by
$$ \begin{equation*} f(x)=\left\{ \begin{array}{ll} \dfrac{1}{\beta - \alpha}, & \hbox{$\alpha \leq x\leq \beta$;} \\ 0, & \hbox{Otherwise.} \end{array} \right. \end{equation*} $$
Distribution Function
The distribution function of continuous uniform distribution $U(\alpha,\beta)$ is
$F(x)=\dfrac{x-\alpha}{\beta - \alpha},\;\alpha \leq x\leq \beta$
Mean of Uniform Distribution
The mean of continuous uniform distribution is
$E(X) = \dfrac{\alpha+\beta}{2}$.
Variance of Uniform Distribution
The variance of continuous uniform distribution is
$V(X) = \dfrac{(\beta - \alpha)^2}{2}$.
Mean deviation about mean of Uniform Distribution
The mean deviation about mean of continuous Uniform Distribution is
$E[|X-\mu_1^\prime|] = \dfrac{\beta-\alpha}{4}$
M.G.F. of Uniform Distribution
The moment generating function of continuous uniform distribution is