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 1x2.

a. (Jensen’s inequality) Suppose that g 1x2 is a concave function. Show that E 3 g 1x2 4 # g 3E 1x2 4 . Hint: Construct the tangent to g 1x2 at the point E 1x2. This tangent will have the form c 1 dx $ g 1x2 for all values of x and c 1 dE 1x2 5 g 3E 1x2 4, where c and d are constants.

b. Use the procedure from part

(a) to show that if g 1x2 is a convex function, then E 3g 1x2 4 $ g 3E 1x2 4.

c. Suppose x takes on only non-negative values—that is, 0 # x # q. Use integration by parts to show that E 1x2 5 3 q

0 31 2 F 1x2 4dx,

where F 1x2 is the cumulative distribution function for x [i.e., F 1x2 5 ex 0 f 1t2dt].

d. (Markov’s inequality) Show that if x takes on only positive values, then the following inequality holds:
P 1x $ t2 #
E 1x2 t .
Hint: E 1x2 5 e q 0 xf 1x2 dx 5 et 0 xf 1x2 dx 1 e q t xf 1x2 dx.

e. Consider the PDF f 1x2 5 2x23 for x $ 1.
1. Show that this is a proper PDF.
2. Calculate F 1x2 for this PDF.
3. Use the results of part

(c) to calculate E 1x2 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 1x0A2. To compute this value we need to know the PDF for x given that A has occurred [denoted by f 1x0A2].
With this notation, E 1x0A2 5 e 1q 2q xf 1x0A2dx. Perhaps the easiest way to understand these relationships is with an example. Let f 1x2 5 x2 3 for 21 # x # 2.
1. Show that this is a proper PDF.
2. Calculate E 1x2.
3. Calculate the probability that 21 # x # 0.
4. Consider the event 0 # x # 2, and call this event A.
What is f 1x0A2?
5. Calculate E 1x0A2.
6. Explain your results intuitively.