Question: Problem 4.59:

(a) Show that the conditional distribution function if the continuous random variable X , given \(a<X\le b\) is given by

\[F(x|a<X\le b)=\left\{ \begin{aligned} & 0\text{ for }x\le a \\ & \frac{F(x)-F(a)}{F(b)-F(a)}\text{ for }a<x\le b \\ & 1\text{ for }x>b \\ \end{aligned} \right.\]

(b) Differentiate the above result with respect to x to find the conditional density of \(X\) given \(a<X\le b\), and show that

\[E(u(X)|a<X\le b)=\frac{\int\limits_{a}^{b}{u(x)f(x)dx}}{\int\limits_{a}^{b}{f(x)dx}}\]