2.14 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 PDF f(x).

a. ( Jensen’s inequality) Suppose that g (x) is a concave function. Show that E[g (x)] - g[E(x)]. Hint: Construct the tangent to g (x) at the point E(x). This tangent will have the form c þ dx + g (x) for all values of x and c þ dE(x) ¼ g[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 E[ g (x)] + g[E(x)].

c. Suppose x takes on only non-negative values—that is, 0 - x - 1. Use integration by parts to show that EðxÞ ¼

1ð

0

½1 % FðxÞ( dx, where F(x) is the cumulative distribution function for x [that is, F(x) ¼ Ð x 0 fðtÞdt(.

d. (Markov’s inequality) Show that if x takes on only positive values then the following inequality holds:

Pðx + tÞ - EðxÞ

t :

Hint: EðxÞ ¼ Ð 1 0 xfðxÞ dx ¼ Ðt 0 xfðxÞ dx þ Ð 1 t xfðxÞ dx.