S&DS 241 / 541 and Math 241: Final Exam (Dec. 17, 2019)
• Please turn in these exam questions along with your blue book.
• Please write your name and NetID (if you have one) on your blue book.
• There are five questions (100 points total). Please show your steps. Good luck!
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Question 1 (35 points)
Suppose X and Y are random variables with the following joint PDF, where c is a constant:
fX ,Y (x,y) =
{
c(x+ y) if 0 < x< y< 1
0 otherwise
(a) Find the value of the constant c.
(b) Find the PDF of Y .
(c) Find the variance of Y . (It’s okay to leave your answer as a difference of fractions.)
(d) Find the conditional PDF of X , given Y = y. (Assume 0 < y< 1.)
(e) Suppose U has the Uniform(0,1) distribution. Find a function g such that g(U) has the same
distribution as Y .
Question 2 (20 points)
Let M and F be the midterm and final exam scores of a random student in Astrology 241.
Suppose (M, F) is Bivariate Normal with E(M) = 50, Var(M) = 9, E(F) = 60, Var(F) = 16,
and Corr(M, F) = 0.5.
(a) Find P(52 < F < 68). Express your answer in terms of Φ (the standard Normal CDF).
(b) Find the distribution of F−M (the name of the distribution and any parameters).
(c) Suppose the student’s homework score H is Normal and independent of (M,F).
Can we conclude that (M, F, H) is Multivariate Normal? Why or why not?
Question 3 (15 points)
(a) Suppose T follows the Exponential(1) distribution, so that its PDF is fT (t) = e− t for t > 0
(and zero otherwise). Find the PDF of V = 1/T .
(b) Suppose U follows the Uniform(−1,1) distribution. Find the PDF of X =U2.
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Question 4 (10 points)
Suppose X follows the Poisson(1) distribution. The probability mass function of X is
P(X = k) =
e−1
k!
for k = 0,1,2, . . .
(and zero otherwise). The moment generating function of X is
MX(t) = e(e
t−1) for all real numbers t.
(We can also write MX(t) = exp(e t−1), which means the same thing.)
(a) Use the moment generating function of X to find E(X) and Var(X).
(b) Find E(X | X > 0). (Hint: Use the Law of Total Expectation.)
Question 5 (20 points)
Suppose U1,U2, . . . ,U400 are i.i.d. Uniform(0,1). The variance of this distribution is 1/12.
(a) Let U = (U1 + . . .+U400)/400. Give an approximate distribution for U (the name of the
distribution and any parameters).
(b) Let Y = min(U1,U2). Find the PDF of Y .
(c) Let V =U1 and W = 2U2. Find the PDF of T =V +W .
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