Let X be a continuous random variable, with CDF F(x), taking values in an interval [0, b]; that is, F(0) = 0 and F(b) = 1. Then there is an...

4. Let X be a continuous random variable, with CDF F(x), taking values in an interval[0, b]; that is, F(0) = 0 and F(b) = 1. Then there is an alternative formula forexpected value:E(X) =Z b0(1 − F(x)) dx. (1)(a) Assume b is a finite number. Prove (1) using integration-by-parts. [Hint:Recall that the PDF is f(x) = ddxF(x).](b)Check that the formula (1) holds when X Unif(0, b).(c) Formula (1) also works for b = 1. Check this when X is an Exponential RVwith PDF f(x) = e−x for x 0.