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

$M_X(t) = \dfrac{e^{t\beta}-e^{t\alpha}}{t(\beta-\alpha)}$

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