1. The following data comes from the 2011 Behavioral Risk Factor Surveillance System (BRFSS) survey, which is run by the Centers for Disease Control (CDC). This is a joint probability table for the proportions of survey respondents who smoke and who have had heart attacks.

Answer the following questions:

(a) What is the proportion of people who smoke?

(b) What is the proportion of people who have had a heart attack?

(c) If a person is a smoker, are they more likely to have had a heart attack than someone who is not a smoker? Hint: Be careful what this question is asking!

(d) If someone tells you that they have had a heart attack, what is the probability that they are also a smoker?

(e) Is smoking independent of having a heart attack?

(f) Define Bernoulli random variables, S = 0 (non-smoker), S = 1 (smoker), and H = 0 (no heart attack), H = 1 (heart attack). What is Cov(S, H)?

(g) What is ρ(S, H)?

2. Let k be some constant number, and consider continuous random variables X and Y with joint pdf 

(a) Find k.

(b) What is the joint probability P

(c) What is the marginal pdf fX(x)?

(d) What is the marginal pdf fY (y)?

(e) What is the conditional probability P(x≤Π/4 ly=Π/4)

(f) Are X and Y independent? Explain why or why not.

(g) What is Cov(X, Y )?

(h) What is ρ(X, Y )?

3. A polling station in Salt Lake City has 50 registered Republicans and 50 registered Democrats. After 4 people show up to vote, let R be the number of Republicans and D be the number of Democrats that have voted.

(a) Compute and list all the entries of the joint probability table P(R = a, D = b) that are non-zero.

(b) What is Cov(R, D)?

(c) What is ρ(R, D)?

Hint: These computations may seem daunting, but the final answers should come out to be simple numbers. You can use a calculator of course, but don’t just give the final answer –show your work!