SID:
1. Consider independent U, V ∼ Exp(3). Let X = min{U, V }, Y = max{U, V }.
(a) Name the distribution of X and give its parameter; (4 points)
(b) Are X and Y independent? Explain briefly; (3 points)
(c) Find the distribution of Y −X. (3 points)
2. Let X and Y have the joint the density fX,Y (x, y) = 180x
2(1 − y)2, 0 < x < y < 1
and 0 otherwise. Your answers should not include any region where the density is
0.
(a) Find the marginal density of Y . Identify the distribution and state the param-
eters. You must show your work. (4 points)
(b) Evaluate P (Y > 3X)? Hint: what distribution is X/Y? (4 points)
For the following part, please only set up the problem and do not solve it ie.
leave your final answer in terms of integrals. Your integrals should not include
any region where the density is 0. Make sure your order of integration is clear.
(c) What is E[eXY ]? (2 points)
3. Darts are thrown at a board, with the X and Y coordinates of the dart following a
Standard Normal Distribution. The center of the board is placed at (0,0).
(a) Find the probability that the dart is between 3 to 5 units away from the origin.
(2 points)
(b) After throwing 4 darts, find the distribution of the minimum distance a dart
is from the origin. Note: If this isn’t a know distribution give the density. (4
points)
(c) A friend offers you $150 if the minimum distance of the 4 darts that you throw
is within 0.5 units of the origin. However, to make an attempt, you must pay
$100. What is your expected profit if you make an attempt? (4 points)
4. Let Tr be a Negative Binomial(r, p) random variable, whose PMF is given by
P (Tr = k) =
(
k + r − 1
k
)
pr(1− p)k, k = 0, 1, 2, · · · ,
which is equal in distribution to the sum of r i.i.d. copies of T1.
(a) Show that the moment generating function MT1(s) = E(e
sT1) of T1 for r = 1
is p
1−(1−p)es for (1− p)e
s < 1. Hint: use a geometric sum. (2 points) (b) Find the MGF of Tr. (3 points) The next part of this question is independent of the first two parts. 1 SID: (c) Suppose a random variable X has the moment generating function MX(t) := E[etX ]. Let g(t) := logMX(t), then prove that g′′(0) = V ar[X]. (5 points) 5. We are interested in tracking the growth of two countries over time. Suppose coun- tries A and B have populations a, b in 2000. By 2100, we model the population of A,B as X = aeZA , Y = beZB , for ZA, ZB i.i.d.∼ N (0, 1). (a) What is the distribution of ZA − ZB? Explain fully. (4 points) (b) Find the density of W = X/Y ? (6 points) 2