.1. Let X and Y be random variables taking the values on the set 0 ? X ? Y with joint density functionfX,Y (x, y) = (4e?3xe?yfor 0 ? x ? y0 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 ybut 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. Eachtime I receive an email, the probability that it is spam is 0.3, independent of all other emails. Thusif I receive N emails, the number of emails which are spam is a Bin(N, 0.3) random variable. Let Nbe the number of emails I will receive tonight, and let X be the number of spam emails I will receivetonight.(a) Use “double expectation” to compute the expected value of the number of spam emails I willreceive tonight, E(X).(b) Use “double expectation” to compute the variance of the number of spam emails I will receivetonight, 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)), sincewhen conditioning on N, we are thinking of N as a constant. You will have to use the fact thatE(N2) = E(N)2+Var(N).(e) Use your answers to parts (b) and (d) to find the correlation coefficient between X and N.. A recent NYT/Sienna poll showed that in the upcoming presidential election, 36% of Americans planto 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 gather5 randomly selected Americans in a room. Find the probability that there are the same number ofTrump 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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