692. Find an explicit formula for the simulation of a continuous random variable given by the probability density function $f(x)=b /(1+a x)^{2}$ in the interval $[0,1 /(b-a)]$; outside this interval $f(x)=0$.
Solution. Using rule 2, we write the equation

$$
b \int_{0}^{x_{i}} 1 /(1+a x)^{2} \mathrm{~d} x=r_{i}
$$

Solving this equation for $x_{i}$, we finally obtain

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