## 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