Q1. Alice, Bob, and Carl arrange to meet for lunch on a certain day. They arrive independently at uniformly distributed times between 1 pm and 1:30 pm on that day.
(a) What is the probability that Carl arrives first? For the rest of this problem, assume that Carl arrives rst at 1:10 pm, and condition on this fact.
(b) What is the probability that Carl will have to wait more than 10 minutes for one of the others to show up? (So consider Carl's waiting time until at least one of theothers has arrived.)
(c) What is the probability that Carl will have to wait more than 10 minutes for both of the others to show up? (So consider Carl's waiting time until both of the others has arrived.)
(d) What is the probability that the person who arrives second will have to wait more than 5 minutes for the third person to show up?
Q2. A stick of length L (a positive constant) is broken at a uniformly random point X. Given that X = x, another breakpoint Y is chosen uniformly on the interval [0; x].
(a) Find the joint PDF of X and Y . Be sure to specify the support.
(b) We already know that the marginal distribution of X is Unif(0; L). Check that marginalizing out Y from the joint PDF agrees that this is the marginal distribution of X.
(c) We already know that the conditional distribution of Y given X = x is Unif(0; x).
Check that using the de nition of conditional PDFs (in terms of joint and marginal PDFs) agrees that this is the conditional distribution of Y given X = x.
(d) Find the marginal PDF of Y .
(e) Find the conditional PDF of X given Y = y.
Q3.
(a) Five cards are randomly chosen from a standard deck, one at a time with replacement. Let X; Y;Z be the numbers of chosen queens, kings, and other cards. Find the joint PMF of X; Y;Z.
(b) Find the joint PMF of X and Y .
Hint: In summing the joint PMF of X; Y;Z over the possible values of Z, note that most terms are 0 because of the constraint that the number of chosen cards is ve.
(c) Now assume instead that the sampling is without replacement (all 5-card hands are equally likely). Find the joint PMF of X; Y;Z.
Hint: Use the naive de nition of probability.
Q4. Each of n 2 people puts his or her name on a slip of paper (no two have the same name). The slips of paper are shued in a hat, and then each person draws one (uniformly at random at each stage, without replacement). Find the standard deviation of the number of people who draw their own names.
Q5. You are playing an exciting game of Battleship. Your opponent secretly positions ships on a 10 by 10 grid and you try to guess where the ships are. Each of your guesses is a hit if there is a ship there and a miss otherwise. The game has just started and your opponent has 3 ships: a battleship (length 4), a submarine (length 3), and a destroyer (length 2). (Usually there are 5 ships to start, but to simplify the calculations we are considering 3 here.) You are playing a variation in which you unleash a salvo, making 5 simultaneous guesses. Assume that your 5 guesses are a simple random sample drawn from the 100 grid positions.
Find the mean and variance of the number of distinct ships you will hit in your salvo. (Give exact answers in terms of binomial coeffcients or factorials, and also numerical values computed using a computer.)
Hint: First work in terms of the number of ships missed, expressing this as a sum of indicator r.v.s. Then use the fundamental bridge and naive de nition of probability, which can be applied since all sets of 5 grid positions are equally likely.
Q6. The number of people who visit the Leftorium store in a day is Pois(100). Suppose that 10% of customers are sinister (left-handed), and 90% are dexterous (right-handed). Half of the sinister customers make purchases, but only a third of the dexterous customers make purchases. The characteristics and behavior of people are independent, with probabilities as described in the previous two sentences. On a certain day, there are 42 people who arrive at the store but leave without making a purchase. Given this information, what is the conditional PMF of the number of customers on that day who make a purchase?
Q7. There will be X Pois() courses o ered at a certain school next year.
(a) Find the expected number of choices of 4 courses (in terms of , fully simpli ed), assuming that simultaneous enrollment is allowed if there are time conflicts.
(b) Now suppose that simultaneous enrollment is not allowed. Suppose that most faculty only want to teach on Tuesdays and Thursdays, and most students only want to take courses that start at 10 am or later, and as a result there are only four possible time slots: 10 am, 11:30 am, 1 pm, 2:30 pm (each course meets Tuesday-Thursday for an hour and a half, starting at one of these times). Rather than trying to avoid major con icts, the school schedules the courses completely randomly: after the list of courses for next year is determined, they randomly get assigned to time slots, independently and with probability 1=4 for each time slot. Let Xam and Xpm be the number of morning and afternoon courses for next year, respectively (where \morning" means starting before noon). Find the joint PMF of Xam and Xpm, i.e., nd P(Xam = a;Xpm = b) for all a; b.
(c) Continuing as in (b), let X1;X2;X3;X4 be the number of 10 am, 11:30 am, 1 pm, 2:30 pm courses for next year, respectively. What is the joint distribution of X1;X2;X3;X4? (The result is completely analogous to that of Xam;Xpm; you can derive it by thinking conditionally, but for this part you are also allowed to just use the fact that the result is analogous to that of (b).) Use this to nd the expected number of choices of 4 non-con icting courses (in terms of , fully simpli ed). What is the ratio of the expected value from (a) to this expected value?
Q8. Let X be the number of statistics majors in a certain college in the Class of 2030, viewed as an r.v. Each statistics major chooses between two tracks: a general track in statistical principles and methods, and a track in quantitative nance. Suppose that each statistics major chooses randomly which of these two tracks to follow, independently, with probability p of choosing the general track. Let Y be the number of statistics majors who choose the general track, and Z be the number of statistics majors who choose the quantitative nance track.
(a) Suppose that X Pois(). (This isn't the exact distribution in reality since a Poisson is unbounded, but it may be a very good approximation.) Find the correla- tion between X and Y .
(b) Let n be the size of the Class of 2030, where n is a known constant. For this part and the next, instead of assuming that X is Poisson, assume that each of the n students chooses to be a statistics major with probability r, independently. Find the joint distribution of Y , Z, and the number of non-statistics majors, and their marginal distributions.
(c) Continuing as in (b), nd the correlation between X and Y .
Q9. Let X and Y be i.i.d. N(0; 1) r.v.s, and (R; ) be the polar coordinates for the point (X; Y ), so X = Rcos and Y = Rsin with R 0 and 2 [0; 2). Find the joint PDF of R2 and . Also nd the marginal distributions of R2 and , giving their names (and parameters) if they are distributions we have studied before.
Q10. Alice walks into a post o ce with 2 clerks. Both clerks are in the midst of serving customers, but Alice is next in line. The clerk on the left takes an Expo(1) time to serve a customer, and the clerk on the right takes an Expo(2) time to serve a customer. Let T1 be the time until the clerk on the left is done serving his or her current customer, and de ne T2 likewise for the clerk on the right.
(a) If 1 = 2, is T1=T2 independent of T1 + T2?
Hint: T1=T2 = (T1=(T1 + T2))=(T2=(T1 + T2)):
(b) Find P(T1 < T2) (do not assume 1 = 2 here or in the next part, but do check that your answers make sense in that special case).
(c) Find the expected total amount of time that Alice spends in the post o ce (assuming that she leaves immediately after she is done being served).
Q11. An engineer is studying the reliability of a product by performing a sequence of n trials. Reliability is de ned as the probability of success. In each trial, the product succeeds with probability p and fails with probability 1 p. The trials are conditionally independent given p. Here p is unknown (else the study would be unnecessary!). The engineer takes a Bayesian approach, with p Unif(0; 1) as prior. Let r be a desired reliability level and c be the corresponding con dence level, in the sense that, given the data, the probability is c that the true reliability p is at least r. For example, if r = 0:9; c = 0:95, we can be 95% sure, given the data, that the product is at least 90% reliable. Suppose that it is observed that the product succeeds all n times. Find a simple equation for c as a function of r.
Q12. We are about to observe random variables Y1; Y2; : : : ; Yn, i.i.d. from a continuous distribution. We will need to predict an independent future observation Ynew, which will also have the same distribution. The distribution is unknown, so we will construct our prediction using Y1; Y2; : : : ; Yn rather than the distribution of Ynew. In forming a prediction, we do not want to report only a single number; rather, we want to give a predictive interval with \high con dence" of containing Ynew. One approach to this is via order statistics.
(a) For xed j and k with 1 j < k n, nd P(Ynew 2 [Y(j); Y(k)]): Hint: By symmetry, all orderings of Y1; : : : ; Yn; Ynew are equally likely.
(b) Let n = 99. Construct a predictive interval, as a function of Y1; : : : ; Yn, such that the probability of the interval containing Ynew is 0:95.