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Derivation of Results in Key Concept 2.3 109

c. Let Xn = g(Z) denote another guess of X using Z, and V = X - Xn
denote its error. Show that E(V2) Ú E(W 2). [Hint: Let h(Z ) =
g(Z) - E(X ͉ Z), so that V = 3X - E(X ͉ Z )4 - h(Z ). Derive
E(V2).]

Empirical Exercise

E2.1 On the text website, http://www.pearsonglobaleditions.com/Stock_Watson,
you will find the spreadsheet Age_HourlyEarnings, which contains the
joint distribution of age (Age) and average hourly earnings (AHE) for
25- to 34-year-old full-time workers in 2012 with an education level that
exceeds a high school diploma. Use this joint distribution to carry out the
following exercises. (Note: For these exercises, you need to be able to carry
out calculations and construct charts using a spreadsheet.)

a. Compute the marginal distribution of Age.
b. Compute the mean of AHE for each value of Age; that is, compute,

E(AHE|Age = 25), and so forth.
c. Compute and plot the mean of AHE versus Age. Are average hourly

earnings and age related? Explain.
d. Use the law of iterated expectations to compute the mean of AHE;

that is, compute E(AHE).
e. Compute the variance of AHE.
f. Compute the covariance between AHE and Age.
g. Compute the correlation between AHE and Age.
h. Relate your answers in parts (f) and (g) to the plot you constructed

in (c).

Appendix

2.1 Derivation of Results in Key Concept 2.3

This appendix derives the equations in Key Concept 2.3.
Equation (2.29) follows from the definition of the expectation.
To derive Equation (2.30), use the definition of the variance to write var(a + bY) =

E{[a + bY - E(a + bY)]2} = E{[b(Y - mY)]2} = b2E[(Y - mY)2] = b2s2Y.