1. Please write the following statement “”I have not given or received any unauthorized assistance on
this examination” and sign it. Your work will not be graded without such a signed statement.
2. For the Calculations questions, be sure to show your computations and to write down whatever
equation you use. In order to receive full credit for a problem, you should show all of your work
and explain your reasoning. Circle your final answer. Good work can receive substantial partial
credit even if the final answer is incorrect.
3. If you make a mistake in a multi-part question, later parts will be graded for following through
correctly. You should try all parts of a problem even if you get stuck on an early part. If necessary,
state and assume a value from an earlier part.
4. If you are unsure of what a question is asking for, do not hesitate to ask the instructor for clarification
via email.
5. You may need to use a calculator to answer some questions.
Points Value
Multiple Choice 20
Question 5 16
Question 6 16
Question 7 12
Question 8 16
Question 9 10
Question 10 10
TOTAL 100
1 Multiple Choice (20 points)
Do not explain your answers for this section—just circle one.
1. [5 pts] Let A and B be events, which each have nonzero probability. Suppose that A and B are
disjoint. Are A and B independent?
(a) Yes. With the information given, we can conclude that A and B are independent.
(b) No. With the information given, we can conclude that A and B are not independent.
(c) Unknown. There is not enough information given to determine whether A and B are indepen-
dent or not.
.
2. [5 pts] For independent random variables X and Y
Var(X − Y ) = Var(X)− Var(Y )
is:
(a) True
(b) False
.
3. [5 pts] For a continuous random variable X with density f(x),
E(log(X)) =
∫ ∞
x=−∞
log(x) · f(log(x)) dx.
(a) True
(b) False
.
4. [5 pts] Suppose we draw 10 cards from a standard deck, without replacement. Then the distribution
of X, the number of Kings that we have, is Binomial(10, 4/52).
(a) True
(b) False
(Note—you do not need to numerically calculate the correlation/covariance to answer this question.)
.
2 Calculations (80 points)
For full credit, show your work for all problems in this section.
5. [16 pts] Suppose I roll a die 10 times. Let X = # of 1’s and Y = # of 2’s.
(a) [8 pts] Calculate P(X = 5 | Y = 5).
(b) [8 pts]Calculate E(XY )
6. [16 pts] Let X, Y both be Exponential(2) random variables, X, Y are independent. This distribu-
tion has density fX(x) = 2e
−2x, fY (y) = 2e−2y and CDF FX(x) = 1 − e−2x, FY (y) = 1 − e−2y on
the support x, y ∈ (0,∞).
(a) [6 pts] Find the PDF of X − Y .
(b) [6 pts] Let Z = bXc, which is X rounded down to an integer. For example if X = 2.1 then
Z = 2, or if X = 5.9 then Z = 5. Find the CDF of Y − Z. You can have answers with
summation/integral notations.
(c) [4 pts] Let D = X − Z, the decimal part of X. For example if X = 2.1 then Z = 2 and
D = 0.1. Are Z and D independent or not? Explain.
7. [12 pts] Consider the following procedure. We have a fair coin. First, we flip the coin once. Let
Y = the outcome of the first flip (Y = 1 if Heads, Y = 0 if Tails). If the first flip comes up Heads,
we then flip the coin two additional times; if the first flip comes up Tails, then we instead flip the
coin four additional times. Let Z be the number of Heads in these additional flips (not counting the
initial first flip).
(a) [6 pts] Calculate E(Z).
(b) [6 pts] Calculate P(Y = 0 | Z = 0).
8. [16 pts] Let X and Z be independent draws from a Uniform[−1, 1] distribution.
(a) [6 pts] Let Y = 1 − X2. Find either the CDF of Y , FY (y), or the density of Y , fY (y), and
also state the support of Y , i.e. the range of possible values. You do not need to compute both
the CDF and the density.
(b) [4 pts] Multiple choice—what is the correlation, Corr(X,Y )? Circle one:
i. Equal to −1.
ii. Negative, but not −1.
iii. Equal to 0.
iv. Positive, but not 1.
v. Equal to 1.
You do not need to calculate the correlation, but you may briefly explain your answer (not
required).
(c) [6 pts] Let A = (X +Z)2 and calculate E(A). You may use the following facts: a Uniform[0,1]
random variable U has E(U) = 12 and Var(U) =
1
12 .
9. [10 pts] Let (X,Y ) be random variables supported on the unit square [0, 1]×[0, 1], with joint density
f(x, y) = 0.5 + 2xy. Calculate the conditional density fY |X(y | x).
10. [10 pts] The unit interval [0, 1] is broken into two pieces, with the break point chosen uniformly
at random along the interval. Next, a point X on the unit interval [0, 1] is chosen uniformly at
random. What is the probability that X is in the larger of the two pieces? It’s fine to have integral
or summation notation in your final answer, but no probability/statistics notation such as pX or
E(X) etc.