8. (Devroye, 1986, p. 38) Suppose X is a random variable having cdf F, and Y is a truncated version of this random variable with support restricted to the interval [a, b]. Then Y has cdf G(y) =

⎧

⎨

⎩

0 , y

F (b)−F

(a) , a ≤ y ≤ b 1 , y>b

.

Show that Y can be generated as F −1(F

(a) + U[F

(b) − F(a)]), where U is a Unif(0, 1) random variate. (This result enables “one-for-one”

generation from truncated distributions for which we can compute F and F −1, either exactly or numerically.)