STAT4101 Fall 2007 Practice Exam 2. You are permitted two sheets of paper with notes, frontand back, and a calculator. You will be given a copy of the normal table and the table ofdistributions from the inside covers of your book.1. Let the random variable X have a probability density function (pdf) off(x) =(c(1 − x2) for − 1 < x < 10 otherwise.(a) Find c.(b) Find the cumulative distribution function (cdf) of X, F (x).(c) Find P (X > 0).(d) Find E11−x2.2. Suppose scores on a certain entrance exam are approximately normal with mean 75 andstandard deviation 10.(a) School A sets the cutoff for admission at 68. What percent of students are admitted?(b) Given that a particular student was admitted to school A, what’s the probabilitythat their score was above 75?(c) School B wants to set their cutoff for admission so that only 25% of students areadmitted. What should the cutoff be?3. Suppose the random variable X has moment generating function ofmX(t) = e3t+2t2.(a) Identify the distribution of X.(b) We know that if Y = aX + b, mY(t) = ebmX(at).What is the moment generating function of Y =X2− 3?(c) Identify the distribution of Y .4. Let X and Y have the following joint probability function:Y1 2 3X 1 0.08 0.1 0.222 0.12 0.2 0.28(a) Find P (X = 1, Y ≤ 2).(b) Find the marginal density of X.(c) Find the conditional density of X|Y = 1.(d) Find E(X|Y = 1).(e) Are X and Y independent? Why or why not?15. Let X and Y be uniform over the region0 1 20 1 2xyso it has pdff(x, y) =(12for x > 0, y > 0, x + y < 2,0 otherwise.(a) Find P (Y < 1).(b) Find P (Y < 1|X = 1).(c) Find the marginal density of X.(d) Find the conditional density of Y |X.(e) Find E(Y |X).(f) Are X and Y independent? Why or why not?6. Let X and Y have joint densityf(x, y) = cxy + 1where c is the necessary constant.(a) Find the marginal density of Y (in terms of c).(b) Find the conditional density of X|Y (in terms of c).(c) Find E(Y + 1) (in terms of c).(d) Set up the integral to find P (X < Y ).(e) Are X and Y independent? Why or why not?7. Let E(X) = 5, Var X = 4, E(Y ) = 3, Var Y = 1, and E(XY ) = 16.(a) Find the correlation between X and Y .(b) Find E(2X − 3Y + 4).(c) Find Var(2X − 3Y + 4).8. Let X|Y be Binomial with n = 100 and p = Y , and let Y be Beta with α = 99 and β = 1.Find