Statistics

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.

(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).
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