## 4.2.1 Uniform Distribution

We have already seen the uniform distribution. In particular, we have the following definition:

A continuous random variable $X$ is said to have a *Uniform* distribution over the interval $[a,b]$,
shown as $X \sim Uniform(a,b)$, if its PDF is given by
\begin{equation}
\nonumber f_X(x) = \left\{
\begin{array}{l l}
\frac{1}{b-a} & \quad a < x < b\\
0 & \quad x < a \textrm{ or } x > b
\end{array} \right.
\end{equation}

We have already found the CDF and the expected value of the uniform distribution. In particular, we know that if $X \sim Uniform(a,b)$, then its CDF is given by equation 4.1 in example 4.1, and its mean is given by $$EX=\frac{a+b}{2}$$ To find the variance, we can find $EX^2$ using LOTUS:

$EX^2$ | $= \int_{-\infty}^{\infty} x^2f_X(x)dx$ |

$=\int_{a}^{b} x^2 \left(\frac{1}{b-a}\right) dx$ | |

$=\frac{a^2+ab+b^2}{3}$. |

Therefore,

$\textrm{Var}(X)$ | $=EX^2-(EX)^2$ |

$=\frac{(b-a)^2}{12}$. |