[8 points] Two fair, six-sided die are rolled. Compute the probability that the sum of the

outcomes of the two rolls is:

(a) (3 pt) Equal to 1

(b) (3 pt) Equal to 4

(c) (2 pt) Less than 13

2. [15 points] Suppose that X,Y are discrete random variables with joint probability

(a) (3 pt) Compute the conditional distributions p(X|Y = y) for all possible values of y.

(b) (2 pt) Compute the conditional distributions p(Y|X = x) for all possible values of x.

(c) (3 pt) Compute E[X|Y = y] for all values of y.

(d) (2 pt) Plot E[X|Y = y] against y.

(e) (5 pt) Are X and Y dependent? If yes, explain why. If no, how do you think X and Y

are related?

3. [20 points] You have a large jar containing 999 fair coins and one two-headed coin (i.e.

a coin that is guaranteed to come up heads). Suppose you pick one coin at random out of

the jar and flip it 10 times.

(a) (2 pt) What is the probability you selected a fair coin?

(b) (2 pt) What is the probability of the first flip being heads, assuming you selected a

fair coin?

(c) (2 pt) What is the probability of the first flip being heads, assuming you selected the

two-headed coin?

(d) (3 pt) What is the overall probability of the first flip being heads?

(e) (3 pt) What is the probability of the first and second flip being heads, assuming you

selected a fair coin?

(f) (8 pt) What is the probability you selected a fair coin if all ten flips turn up heads?

4. [32 points] A kidnapping recently occurred in the city of Twin Peaks. Dale, an FBI

agent, has been assigned to catch the person who did it. He has determined that the

guilty person must be one of the 1,000 members of a club called the White Lodge.

(a) (2 pt) Leland is a member of the White Lodge. Based on only this information,

what is the probability that Leland is guilty?

(b) (6 pt) Dale has an expensive DNA test that he can use to help his investigation.

Suppose the probability of a DNA match given that a person is innocent is 0.1%,

and the probability of a DNA match given that a person is guilty is 99%. What is the

probability of a DNA match for a randomly chosen member of the White Lodge?

(c) (9 pt) Dale has found DNA evidence at the scene of the kidnapping. An analysis

of the DNA matches a sample that Leland provides. What is the probability that

Leland is guilty given the positive outcome of the DNA test? Explain intuitively

why this number is higher or lower than that of part (a).

(d) (12 pt) Bob is trying to sell Dale a lie detector machine that he claims is very

accurate. He says: “If my machine starts beeping, you can be 99% sure the person

really is guilty”. What must be the relative fraction of times the machine starts

beeping for a guilty person, over the times it starts beeping for an innocent person?

(e) (3 pt) Given just the above information, is it possible to determine the fraction of

times that Bob’s machine starts beeping for a guilty person? If yes, compute the

fraction. If no, explain why not.

5. [25 points] Suppose that X is a random variable with values {0,1}, and p(X = 1) = θ.

In the following, express your answer in terms of θ.

(a) (3 pt) Define and calculate the expectation of X.

(b) (4 pt) Define and calculate the variance of X.

(c) (6 pt) Calculate the quantity φ(t) := E[e

tX] for a fixed t > 0.

(d) (6 pt) Obtain an expression for the derivative φ

0

(t). Explain the relation of φ

0

(0) to

the result of (a).

(e) (6 pt) Obtain an expression for the second derivative φ

00(t). Explain the relation of

φ

00(0) to the result of (b).

Read less