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108 Chapter 2 Review of Probability

2.24 Suppose Yi is distributed i.i.d. N(0, s2) for i = 1, 2, c, n.

a. Show that E(Y 2 > s2) = 1.
i

b. Show that W = (1 > s2) g n 1Y 2 is distributed xn2.
i= i

c. Show that E(W) = n. [Hint: Use your answer to (a).]

d. Show that V = Y1 n g n 2Yi2 is distributed tn - 1.
i=

B n-1

2.25 (Review of summation notation) Let x1, c, xn denote a sequence of
numbers, y1, c, yn denote another sequence of numbers, and a, b, and c
denote three constants. Show that

nn

a. ia= 1axi = aia= 1xi

n nn

b. ia= 1(xi + yi) = ia= 1xi + ia= 1yi

n

c. ia= 1a = na

n nn n
d. ia= 1(a + bxi + cyi)2 = na2 + b2ia= 1x2i + c2ia= 1yi2 + 2abia= 1xi +

nn

2acia= 1yi + 2bcia= 1xiyi

2.26 Suppose that Y1, Y2, c, Yn are random variables with a common mean mY,
a common variance sY2, and the same correlation r (so that the correlation
between Yi and Yj is equal to r for all pairs i and j, where i j).

a. Show that cov(Yi, Yj) = rsY2 for i j.
b. Suppose that n = 2. Show that E(Y ) = mY and var(Y ) = 12s2Y + 21rs2Y.
c. For n Ú 2, show that E(Y ) = mY and var(Y ) = sY2 >n +

[(n - 1)>n]rs2Y.
d. When n is very large, show that var(Y ) ≈ rs2Y.

2.27 X and Z are two jointly distributed random variables. Suppose you know

the value of Z, but not the value of X. Let X = E(X ͉ Z) denote a guess
of the value of X using the information on Z, and let W = X - X denote
the error associated with this guess.

a. Show that E(W ) = 0. (Hint: Use the law of iterated expectations.)
b. Show that E(WZ ) = 0.