Page 106 -

Exercises 105

b. If Y is distributed t30, find Pr( - 1.70 … Y … 1.70).
c. If Y is distributed N(0, 1), find Pr( -1.70 … Y … 1.70).
d. When do the critical values of the normal and the t distribution coincide?

e. If Y is distributed F4,11, find Pr(Y 7 3.36).
f. If Y is distributed F3,21, find Pr(Y 7 4.87).

2.13 X is a Bernoulli random variable with Pr(X = 1) = 0.90, Y is distributed
N(0, 4), W is distributed N(0, 16), and X, Y, and W are independent. Let
S = XY + (1 - X)W. (That is, S = Y when X = 1, and S = W when
X = 0.)

a. Show that E(Y 2) = 4 and E(W 2) = 16.
b. Show that E(Y3) = 0 and E(W3) = 0. (Hint: What is the skewness

for a symmetric distribution?)
c. Show that E(Y4) = 3 * 42 and E(W4) = 3 * 162. (Hint: Use the fact

that the kurtosis is 3 for a normal distribution.)
d. Derive E(S), E(S2), E(S3) and E(S4). (Hint: Use the law of iterated

expectations conditioning on X = 0 and X = 1.)
e. Derive the skewness and kurtosis for S.

2.14 In a population mY = 50 and JY2 = 21. Use the central limit theorem to
answer the following questions:

a. In a random sample of size n = 50, find Pr(Y … 51).
b. In a random sample of size n = 150, find Pr(Y 7 49).
c. In a random sample of size n = 45, find Pr(50.5 … Y … 51).

2.15 Suppose Yi, i = 1, 2, c, n, are i.i.d. random variables, each distributed
N(10, 4).

a. Compute Pr(9.6 … Y … 10.4) when (i) n = 20, (ii) n = 100, and
(iii) n = 1000.

b. Suppose c is a positive number. Show that Pr(10 - c … Y … 10 + c)
becomes close to 1.0 as n grows large.

c. Use your answer in (b) to argue that Y converges in probability
to 10.

2.16 Y is distributed N(5, 100) and you want to calculate Pr(Y 6 3.6). Unfor-
tunately, you do not have your textbook, and do not have access to a nor-
mal probability table like Appendix Table 1. However, you do have your