2.11 More on expected value Because the expected value concept plays an important role in many economic theories, it may be useful to summarize a few more properties of this statistical measure. Throughout this problem, x is assumed to be a continuous random variable with probability density function f(x).

a. (Jensen's inequality) Suppose that g(x) is a concave function. Show that Eg(x)] [E(x)]. Hint: Construct the tangent to g(x) at the point E(x). This tangent will have the form +dx g(x) for all values of x and e+dE(x) = E(x)] where c and d are constants.

b. Use the procedure from part

(a) to show that if g(x) is a convex function then Elg(x)] g|E(x)].

c. Suppose x takes on only nonnegative values-that is, 0 x. Use integration by parts to show that E(x) = =(1 F(x) dx, where F(x) is the cumulative distribution function for x [that is, F(x) = f f(t) dt].

d. (Markov's inequality) Show that if x takes on only positive values then the following inequality holds: E(x) P(x2) xf(x)dx. Hint: E(x) = xf(x) dx = Soxf (x) dx +

e. Consider the probability density function f(x) = 2x3 for x 1. (1) Show that this is a proper PDF. (2) Calculate F(x) for this PDF. (3) Use the results of part

(c) to calculate E(x) for this PDF. (4) Show that Markov's inequality holds for this function.

f. The concept of conditional expected value is useful in some economic problems. We denote the expected value of x conditional on the occurrence of some event, A, as E(x|A). To compute this value we need to know the PDF for x given that A has occurred [denoted by f(x4)]. With this notation, E(x|A) fxf(x|A) dx. Perhaps the easiest way to understand these relation- ships is with an example. Let f(x)= - for -1x2. 3 (1) Show that this is a proper PDF. (2) Calculate E(x). (3) Calculate the probability that -1 < x <0. (4) Consider the event 0x2, and call this event A. What is f(x|A)? (5) Calculate E(x|A). (6) Explain your results intuitively.