The number of goals scored during a game by the Toronto Maple Leafs

is a normally distributed random variable X with x = 3 and ?x = 1.2.

The number of goals given up during a game when Curtis Joseph is the

goaltender for the Maple Leafs is a normally distributed random variable Y

with y = 2.85 and ?y = 0.9. Assume that X and Y are independent.

a) What is the probability that the Maple Leafs will win a game in which

Curtis Joseph is the goaltender? (The probability of a game ending in

a tie is zero here.)

b) What is the probability that the Maple Leafs will lose a game by 2 or

more goals when Curtis Joseph is the goaltender?

c) Let T denote the total number of goals scored by both the Maple Leafs

and their opponent during a game in which Curtis Joseph is the Leafs'

goaltender. What is the expected value and variance of T?

d) Given your answer to a) and assuming that the outcomes of consecutive

games are independent, what is the expected number of wins for the

Maple Leafs over 50 games in which Curtis Joseph is the goaltender?

Hint: What kind of process is occurring here?

Which of the following could be quantified as a Bernoulli random variable?

a) number of persons in a hospital ward with terminal diagnoses.

b) weights of deliveries at a supermarket.

c) square foot areas of houses being built in a suburban tract development.

d) whether or not employees wear glasses.

e) none of the above.

9. Fifteen percent of the patients seen in a pediatric clinic have a respiratory

complaint. In a Bernoulli process of 10 patients, what is the probability that

at least three have a respiratory complaint?

a) .1298

b) .1798

c) .1960

d) .9453

e) none of the above.

10. Two random variables X and Y have the following properties: x = 10,

?x = 4, y = 8, ?y = 5, ?x,y = ?12.

a) Find the expected value and variance of (3X ? 4Y ).

b) Find the expected value of X2

. (Hint: work from the definition of the

variance of X.)

c) Find the correlation between X and (X + Y )

2. (50 points) Suppose that a joint probability density function of two random variables, X and Y is given as: fx,x(x, y) = kxey, 0 1). (d) Find the marginal pdf of X. (e) Are X and Y independent? Provide a reasoning. (f) Find the median of X. (g) Find the expected value of X - 2Y. (h) Find the variance of X - 2Y. (i) Find P(X