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106 Chapter 2 Review of Probability

computer and a computer program that can generate i.i.d. draws from the
N(5, 100) distribution. Explain how you can use your computer to compute
an accurate approximation for Pr(Y 6 3.6).

2.17 Yi, i = 1, c, n, are i.i.d. Bernoulli random variables with p = 0.6. Let
Y denote the sample mean.

a. Use the central limit to compute approximations for

i. Pr(Y 7 0.64) when n = 50.

ii. Pr(Y 6 0.56) when n = 200.

b. How large would n need to be to ensure that Pr(0.65 7 Y 7 0.55) = 0.95?
(Hint: Use the central limit theorem to compute an approximate answer.)

2.18 In any year, the weather may cause damages to a home. On a year-to-year

basis, the damage is random. Let Y denote the dollar value of damages in

any given year. Suppose that during 95% of the year Y = $0, but during
the other 5% Y = $30,000.

a. What are the mean and standard deviation of damages caused in a
year?

b. Consider an “insurance pool” of 120 people whose homes are suf-
ficiently dispersed so that, in any year, the damage to different homes
can be viewed as independently distributed random variables. Let Y
denote the average damage caused to these 120 homes in one year.
(i) What is the expected value of the average damage Y? (ii) What is
the probability that Y exceeds $3,000?

2.19 Consider two random variables X and Y. Suppose that Y takes on k values
y1, c, yk and that X takes on l values x1, c, xl.

a. Show that Pr(Y = yj) = g l = 1Pr(Y = yj ͉ X = xi) Pr(X = xi). [Hint:
i

Use the definition of Pr(Y = yj ͉ X = xi).]

b. Use your answer to (a) to verify Equation (2.19).

c. Suppose that X and Y are independent. Show that sXY = 0 and
corr(X, Y) = 0.

2.20 Consider three random variables X, Y, and Z. Suppose that Y takes on

k values y1, c, yk, that X takes on l values x1, c, xl, and that Z takes
on m values z1, c, zm. The joint probability distribution of X, Y, Z is
Pr(X = x, Y = y, Z = z), and the conditional probability distribution of
Pr(Y = y, X = x, Z = z).
Y given X and Z is Pr(Y = y ͉ X = x, Z = z) = Pr(X = x, Z = z)