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102 Chapter 2 Review of Probability

2.3. Suppose that X denotes the amount of rainfall in your hometown during
a randomly selected month and Y denotes the number of children born
in Los Angeles during the same month. Are X and Y independent?
Explain.

2.4. An econometrics class has 80 students, and the mean student weight is
145 lb. A random sample of 4 students is selected from the class, and their
average weight is calculated. Will the average weight of the students in the
sample equal 145 lb? Why or why not? Use this example to explain why
the sample average, Y, is a random variable.

2.5. Suppose that Y1, c, Yn are i.i.d. random variables with a N(1, 4) distri-
bution. Sketch the probability density of Y when n = 2. Repeat this for
n = 10 and n = 100. In words, describe how the densities differ. What is
the relationship between your answer and the law of large numbers?

2.6. Suppose that Y1, c, Yn are i.i.d. random variables with the probability
distribution given in Figure 2.10a. You want to calculate Pr( Y … 0.1).
Would it be reasonable to use the normal approximation if n = 5? What
about n = 25 or n = 100? Explain.

2.7. Y is a random variable with mY = 0, sY = 1, skewness = 0, and
kurtosis = 100. Sketch a hypothetical probability distribution of Y.
Explain why n random variables drawn from this distribution might have
some large outliers.

Exercises

2.1 Let Y denote the number of “heads” that occur when two loaded coins are
tossed. Assume the probability of getting “heads” is 0.4 on either coin.

a. Derive the probability distribution of Y.
b. Derive the mean and variance of Y.

2.2 Use the probability distribution given in Table 2.2 to compute (a) E(Y) and
E(X); (b) s2X and sY2 ; and (c) sXY and corr(X, Y).

2.3 Using the random variables X and Y from Table 2.2, consider two new
random variables W = 4 + 8X and V = 11 - 2Y. Compute (a) E(W) and
E(V); (b) J2W and JV2 ; and (c) JWV and corr(W, V).

2.4 Suppose X is a Bernoulli random variable with P(X = 1) = p.

a. Show E(X3) = p.
b. Show E(Xk) = p for k 7 0.