91Ó°ÊÓ

X has the discrete distribution with cdf \(F\left( x \right)\)

Let \({x_1},{x_2}...\) denote the possible values of X.

Since\(F\left( x \right)\)is a step function, the integral given in exercise 1 that is\(E\left( X \right) = \int\limits_0^\infty {\left( {1 - F\left( x \right)} \right)dx} \)

becomes the following sum:

\(\begin{align}\left( {{x_1} - 0} \right) + \left( {1 - f\left( {{x_1}} \right)} \right)\left( {{x_2} - {x_1}} \right) + \left( {1 - f\left( {{x_1}} \right) - f\left( {{x_2}} \right)} \right)\left( {{x_3} - {x_2}} \right) + ...\\ = {x_1}f\left( {{x_1}} \right) + {x_2}f\left( {{x_2}} \right) + {x_3}f\left( {{x_3}} \right) + ...\\ = E\left( X \right).\end{align}\)

Hence we can say that conclusion of exercise 2 is still holds.