Example 8. The random variable $X$ is uniformly distributed on the interval $[-3,2]$. Find the distribution function $F(x)$ of this random variable.
Solution. The distribution function $F(x)$ of a random variable $X$, uniformly distributed on the interval $[\alpha, \beta]$, is defined by formula (3.4.6). From the problem statement, it follows that $\alpha=-3, \beta=2, \beta-\alpha=2-(-3)=5$; therefore,

$$
F(x)=\left\{\begin{array}{cll}
0 & \text { for } & x \leq-3 \\
\frac{x+3}{5} & \text { for } & -3<x<2 \\
1 & \text { for } & x \geq 2
\end{array}\right.
$$