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#### only three questions, due tomorrow MAT 352, Problem set due Tuesday, June 30, 2020 Please write clear and complete solutions to the following problems, and upload them to Gradescope by class-time on Tuesday, June 30

###### Statistics

only three questions, due tomorrow

MAT 352, Problem set due Tuesday, June 30, 2020 Please write clear and complete solutions to the following problems, and upload them to

Gradescope by class-time on Tuesday, June 30.

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−3xe−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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