[SOLVED] PSTAT 160A Summer 2025 SAMPLE FINAL EXAM

SAMPLE FINAL EXAM

PSTAT 160A– Summer 2025

1. (10 points) A random variable X has continuous distribution with density

(a) Find the moment generating function of X. Make sure to indicate for which t P R the MGF is well-defined.

(b) Let X1, . . . , X5 be five independent and identically distributed (i.i.d.) random variables drawn from the distribution above. If what is ErY
2
s? (Hint: use properties of MGF)

2. (10 points) You generate 10 real numbers X1, . . . , X10 uniformly and independently sampled on (0, 1). Let S : = X1 + … + X10. Without using a computer, estimate the probability that {S → 9}. (There are multiple ways to do it —some are better than others, you will get partial credit for not so good ones).

3. (10 points) Consider a random walk (Sn) that at each step goes up by +2 with probability 0.4, or down -1 with probability 0.6, i.i.d. Note that this walk is NOT simple. There are no boundaries, i.e. the state space are all integers.

Answer TRUE or FALSE: For any i ∈ Z, state i of this stochastic process (Sn) is transient. Provide a paragraph-length explanation of your answer. Be sure to use concepts/terminology we learned in the course.

4. (15 points) Consider continually flipping a three-sided coin, with sides labelled: 1, 2, 3. Let X0 be the outcome of the first roll. Then we recursively define the process as follows: For each subsequent step n = 0, 1, 2, …, we define Xn+1 by the rule

So the state space is S = {0, 1, 2, 3}. Assume that each coin toss is independent of other tosses and that the coin is fair (i.e., that each side of the coin comes up with 1/3 probability).

(a) (4 points) Clearly justify that this is a stationary Markov chain and find the transition matrix P.

(b) (5 points) For each i P S, find Pi(X2 = 0).

(c) (6 points) Find limn→∞ E[Xn]. Explain your solution approach.

5. (15 points) Your professor likes to go on scavenger hunts with his family. There are four zones at his parents’ house: front yard (F), back yard (B), side yard (S), and the deck (D). He moves around every 10 minutes according to the following rules:

• If he is in the side yard, he is equally likely to go to the front, back, or deck.

• If he is in the front yard, he goes to the side yard with probability 1/4, and stays in the front yard with probability 3/4.

• If he is in the back yard, he has a 1/2 probability of staying there. Otherwise, he chooses between the deck and the side yard with equal probability.

• If he goes on the deck, the scavenger hunt is over because he eats snacks and has a beverage (keeps staying on the deck).

Suppose your professor starts the scavenger hunt in the Front Yard.

(a) (4 points) Find the transition matrix with state space S = {F, B, S, D}.

(b) (11 points) Compute the expected number of minutes until he reaches the deck for the first time. Explain your solution approach.