689. Find the explicit formula for the generation of a continuous random variable $X$, uniformly distributed in the interval ( $a, b$ ), given its distribution function $F(x)=(x-a) /(b-a)(a<x<b)$.
Solution. According to Rule 1, we equate the given distribution function to a random number $r_{i}$ :

$$
\left(x_{i}-a\right) /(b-a)=r_{i}
$$

Solving this equation for $x_{i}$, we obtain the explicit formula for generating possible values of $X: x_{i}=(b-a) r_{i}+a$.