1. Let X and Y be random variables taking the values on the set 0 ≤ X ≤ Y with joint density function

fX,Y (x, y) = (

4e

−3x

e

−y

for 0 ≤ x ≤ y

0 otherwise.

(a) Draw the support of the joint density function. Are X and Y independent? Why or why not?

(b) Find the marginal density function for Y .

(c) Write a formula for the conditional density of X given that Y = y for an arbitrary value of y,

fX|Y (x|y).

(d) Compute the conditional expectation E(X|Y = y). The result should depend on the value of y

but not x.

(e) Based on your answer to part (d), do you think the covariance of X and Y should be positive,

negative, or zero? Explain.

2. The number of emails I will receive as I sleep tonight is a Poisson random variable with mean 30. Each

time I receive an email, the probability that it is spam is 0.3, independent of all other emails. Thus

if I receive N emails, the number of emails which are spam is a Bin(N, 0.3) random variable. Let N

be the number of emails I will receive tonight, and let X be the number of spam emails I will receive

tonight.

(a) Use “double expectation” to compute the expected value of the number of spam emails I will

receive tonight, E(X).

(b) Use “double expectation” to compute the variance of the number of spam emails I will receive

tonight, Var(X). You may need to use the fact that E(N2

) = E(N)

2+Var(N).

(c) Without doing any computations, do you expect the covariance of X and N to be positive,

negative, or zero? Explain why.

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(d) Use “double expectation” and your answer to part (a) to compute the covariance of N and X.

Hint: The covariance is E(XN)−E(X)E(N). Since E(N) = 30 and you found E(X) in part (a),

you just need to find E(XN). Conditioning on N, this is E(E(XN|N)) = E(NE(X|N)), since

when conditioning on N, we are thinking of N as a constant. You will have to use the fact that

E(N2

) = E(N)

2+Var(N).

(e) Use your answers to parts (b) and (d) to find the correlation coefficient between X and N.

3. A recent NYT/Sienna poll showed that in the upcoming presidential election, 36% of Americans plan

to support President Trump, 50% plan to support the presumptive Democrat nominee, Joe Biden,

and the remaining 14% are undecided. Assume these numbers are correct, and suppose we gather

5 randomly selected Americans in a room. Find the probability that there are the same number of

Trump supporters as Biden supporters in the room. That is, if T is the number of Trump supporters,

and B is the number of Biden supporters, find P(T = B).

1

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