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,
| $\textrm{Var}(X)$ | $=EX^2-(EX)^2$ |
| $=\frac{(b-a)^2}{12}$. |
