Let X1, X2, X3 be independent with Xi Expo(i) (so with possibly dierent rates). Recall

Fred wants to travel from Blotchville to Blissville, and is deciding between 3 options (involving dierent routes or dierent forms of transportation). The jth option would take an average of j hours, with a standard deviation of j hours. Fred randomly chooses between the 3 options, with equal probabilities. Let T be how long it takes for him to get from Blotchville to Blissville. (a) Find E(T). Is it simply (1 + 2 + 3)/3, the average of the expectations? (b) Find Var(T). Is it simply (2 1 + 2 2 + 2 3)/3, the average of the variances.

While Fred is sleeping one night, X legitimate emails and Y spam emails are sent to him. Suppose that X and Y are independent, with X Pois(10) and Y Pois(40). When he wakes up, he observes that he has 30 new emails in his inbox. Given this information, what is the expected value of how many new legitimate emails he has?

A group of 21 women and 14 men are enrolled in a medical study. Each of them has a certain disease with probability p, independently. It is then found (through extremely reliable testing) that exactly 5 of the people have the disease. Given this information, what is the expected number of women who have the disease? Again given this information, what is the expected number of women who have the disease?

X Pois() be the number of times that a random person got arrested in the last 10 years. However, data from police records are being used for the researchers study, and people who were never arrested in the last 10 years do not appear in the records. In other words, the police records have a selection bias: they only contain information on people who have been arrested in the last 10 years. So averaging the numbers of arrests for people in the police records does not directly estimate E(X); it makes more sense to think of the police records as giving us information about the conditional distribution of how many times a person was arrest, given that the person was arrested at least once in the last 10 years. The conditional distribution of X, given that X 1, is called a truncated Poisson distribution (see Exercise 14 for another example of this distribution). (a) Find E(X|X 1) (b) Find Var(X|X 1)

A fair 20-sided die is rolled repeatedly, until a gambler decides to stop. The gambler pays $1 per roll, and receives the amount shown on the die when the gambler stops (e.g., if the die is rolled 7 times and the gambler decides to stop then, with an 18 as the value of the last roll, then the net payo is $18 $7 = $11). Suppose the gambler uses the following strategy: keep rolling until a value of m or greater is obtained, and then stop (where m is a fixed integer between 1 and 20). (a) What is the expected net payo? Hint: The average of consecutive integers a, a + 1,...,a + n is the same as the average of the first and last of these. See the math appendix for more information about series. (b) Use R or other software to find the optimal value of m.

Let X Expo(). Find E(X|X < 1) in two dierent ways: (a) by calculus, working with the conditional PDF of X given X < 1. (b) without calculus, by expanding E(X) using the law of total expectation.

You get to choose between two envelopes, each of which contains a check for some positive amount of money. Unlike in the two-envelope paradox, it is not given that one envelope contains twice as much money as the other envelope. Instead, assume that the two values were generated independently from some distribution on the positive real numbers, with no information given about what that distribution is. After picking an envelope, you can open it and see how much money is inside (call this value x), and then you have the option of switching. As no information has been given about the distribution, it may seem impossible to have better than a 50% chance of picking the better envelope. Intuitively, we may want to switch if x is small and not switch if x is large, but how do we define small and large in the grand scheme of all possible distributions? [The last sentence was a rhetorical question.] Consider the following strategy for deciding whether to switch. Generate a threshold T Expo(1), and switch envelopes if and only if the observed value x is less than the value of T. Show that this strategy succeeds in picking the envelope with more money with probability strictly greater than 1/2. Hint: Let t be the value of T (generated by a random draw from the Expo(1) distribution). First explain why the strategy works very well if t happens to be in between the two envelope values, and does no harm in any case (i.e., there is no case in which the strategy succeeds with probability strictly less than 1/2).

There are two envelopes, each of which has a check for a Unif(0, 1) amount of money, measured in thousands of dollars. The amounts in the two envelopes are independent. You get to choose an envelope and open it, and then you can either keep that amount or switch to the other envelope and get whatever amount is in that envelope. Suppose that you use the following strategy: choose an envelope and open it. If you observe U, then stick with that envelope with probability U, and switch to the other envelope with probability 1 U. (a) Find the probability that you get the larger of the two amounts. (b) Find the expected value of what you will receive.

Suppose n people are bidding on a mystery prize that is up for auction. The bids are to be submitted in secret, and the individual who submits the highest bid wins the prize. Each bidder receives an i.i.d. signal Xi, i = 1,...,n. The value of the prize, V , is defined to be the sum of the individual bidders signals: V = X1 + + Xn. This is known in economics as the wallet game: we can imagine that the n people are bidding on the total amount of money in their wallets, and each persons signal is the amount of money in his or her own wallet. Of course, the wallet is a metaphor; the game can also be used to model company takeovers, where each of two companies bids to take over the other, and a company knows its own value but not the value of the other company. For this problem, assume the Xi are i.i.d. Unif(0, 1). (a) Before receiving her signal, what is bidder 1s unconditional expectation for V ? (b) Conditional on receiving the signal X1 = x1, what is bidder 1s expectation for V ? (c) Suppose each bidder submits a bid equal to his or her conditional expectation for V , i.e., bidder i bids E(V |Xi = xi). Conditional on receiving the signal X1 = x1 and winning the auction, what is bidder 1s expectation for V ? Explain intuitively why this quantity is always less than the quantity calculated in (b).

A coin with probability p of Heads is flipped repeatedly. For (a) and (b), suppose that p is a known constant, with 0 (a) What is the expected number of flips until the pattern HT is observed? (b) What is the expected number of flips until the pattern HH is observed? (c) Now suppose that p is unknown, and that we use a Beta(a, b) prior to reflect our uncertainty about p (where a and b are known constants and are greater than 2). In terms of a and b, find the corresponding answers to (a) and (b) in this setting.

A fair 6-sided die is rolled once. Find the expected number of additional rolls needed to obtain a value at least as large as that of the first roll.

A fair 6-sided die is rolled repeatedly. (a) Find the expected number of rolls needed to get a 1 followed right away by a 2. Hint: Start by conditioning on whether or not the first roll is a 1. (b) Find the expected number of rolls needed to get two consecutive 1s. (c) Let an be the expected number of rolls needed to get the same value n times in a row (i.e., to obtain a streak of n consecutive js for some not-specified-in-advance value of j). Find a recursive formula for an+1 in terms of an. Hint: Divide the time until there are n + 1 consecutive appearances of the same value into two pieces: the time until there are n consecutive appearances, and the rest. (d) Find a simple, explicit formula for an for all n 1. What is a7 (numerically)?

Let X1, X2 be i.i.d., and let X = 1 2 (X1 + X2) be the sample mean. In many statistics problems, it is useful or important to obtain a conditional expectation given X. As an example of this, find E(w1X1 +w2X2|X), where w1, w2 are constants with w1 +w2 = 1.

Let X1, X2,... be i.i.d. r.v.s with mean 0, and let Sn = X1 + + Xn. As shown in Example 9.3.6, the expected value of the first term given the sum of the first n terms is E(X1|Sn) = Sn n . Generalize this result by finding E(Sk|Sn) for all positive integers k and n.

Consider a group of n roommate pairs at a college (so there are 2n students). Each of these 2n students independently decides randomly whether to take a certain course, with probability p of success (where success is defined as taking the course). Let N be the number of students among these 2n who take the course, and let X be the number of roommate pairs where both roommates in the pair take the course. Find E(X) and E(X|N).

Show that E((Y E(Y |X))2|X) = E(Y 2|X) (E(Y |X))2, so these two expressions for Var(Y |X) agree.

Let (Z,W) be Bivariate Normal, constructed as in Example 7.5.10, so Z = X W = X + p1 2Y, with X, Y i.i.d. N (0, 1). Find E(W|Z) and Var(W|Z). Hint for the variance: Adding a constant (or something acting as a constant) does not aect variance.

Let X be the height of a randomly chosen adult man, and Y be his fathers height, where X and Y have been standardized to have mean 0 and standard deviation 1. Suppose that (X, Y ) is Bivariate Normal, with X, Y N (0, 1) and Corr(X, Y ) = . (a) Let y = ax + b be the equation of the best line for predicting Y from X (in the sense of minimizing the mean squared error), e.g., if we were to observe X = 1.3 then we would predict that Y is 1.3a + b. Now suppose that we want to use Y to predict X, rather than using X to predict Y . Give and explain an intuitive guess for what the slope is of the best line for predicting X from Y . (b) Find a constant c (in terms of ) and an r.v. V such that Y = cX + V , with V independent of X. Hint: Start by finding c such that Cov(X, Y cX) = 0. (c) Find a constant d (in terms of ) and an r.v. W such that X = dY + W, with W independent of Y . (d) Find E(Y |X) and E(X|Y ). (e) Reconcile (a) and (d), giving a clear and correct intuitive explanation.

Let X Mult5(n, p). (a) Find E(X1|X2) and Var(X1|X2). (b) Find E(X1|X2 + X3).

Let Y be a discrete r.v., A be an event with 0 < P(A) < 1, and IA be the indicator r.v. for A. (a) Explain precisely how the r.v. E(Y |IA) relates to the numbers E(Y |A) and E(Y |Ac). (b) Show that E(Y |A) = E(Y IA)/P(A), directly from the definitions of expectation and conditional expectation. Hint: First find the PMF of Y IA in terms of P(A) and the conditional PMF of Y given A. (c) Use (b) to give a short proof of the fact that E(Y ) = E(Y |A)P(A)+E(Y |Ac)P(Ac).

Show that the following version of LOTP follows from Adams law: for any event A and continuous r.v. X with PDF fX, P(A) = Z 1 1 P(A|X = x)fX(x)dx.

Let X and Y be random variables with finite variances, and let W = Y E(Y |X). This is a residual: the dierence between the true value of Y and the predicted value of Y based on X. (a) Compute E(W) and E(W|X). (b) Compute Var(W), for the case that W|X N (0, X2) with X N (0, 1).

One of two identical-looking coins is picked from a hat randomly, where one coin has probability p1 of Heads and the other has probability p2 of Heads. Let X be the number of Heads after flipping the chosen coin n times. Find the mean and variance of X.

Kelly makes a series of n bets, each of which she has probability p of winning, independently. Initially, she has x0 dollars. Let Xj be the amount she has immediately after her jth bet is settled. Let f be a constant in (0, 1), called the betting fraction. On each bet, Kelly wagers a fraction f of her wealth, and then she either wins or loses that amount. For example, if her current wealth is $100 and f = 0.25, then she bets $25 and either gains or loses that amount. (A famous choice when p > 1/2 is f = 2p 1, which is known as the Kelly criterion.) Find E(Xn) (in terms of n, p, f, x0). Hint: First find E(Xj+1|Xj )

Let N Pois(1) be the number of movies that will be released next year. Suppose that for each movie the number of tickets sold is Pois(2), independently. Find the mean and variance of the number of movie tickets that will be sold next year.

A party is being held from 8:00 pm to midnight on a certain night, and N Pois() people are going to show up. They will all arrive at uniformly random times while the party is going on, independently of each other and of N. (a) Find the expected time at which the first person arrives, given that at least one person shows up. Give both an exact answer in terms of , measured in minutes after 8:00 pm, and an answer rounded to the nearest minute for = 20, expressed in time notation (e.g., 8:20 pm). (b) Find the expected time at which the last person arrives, given that at least one person shows up. As in (a), give both an exact answer and an answer rounded to the nearest minute for =20.

We wish to estimate an unknown parameter , based on an r.v. X we will get to observe. As in the Bayesian perspective, assume that X and have a joint distribution. Let be the estimator (which is a function of X). Then is said to be unbiased if E(|) = , and is said to be the Bayes procedure if E(|X) = . (a) Let be unbiased. Find E( ) 2 (the average squared dierence between the estimator and the true value of ), in terms of marginal moments of and . Hint: Condition on . (b) Repeat (a), except in this part suppose that is the Bayes procedure rather than assuming that it is unbiased. Hint: Condition on X. (c) Show that it is impossible for to be both the Bayes procedure and unbiased, except in silly problems where we get to know perfectly by observing X. Hint: If Y is a nonnegative r.v. with mean 0, then P(Y = 0) = 1.

Show that if E(Y |X) = c is a constant, then X and Y are uncorrelated. Hint: Use Adams law to find E(Y ) and E(XY ).

Show by example that it is possible to have uncorrelated X and Y such that E(Y |X) is not a constant. Hint: Consider a standard Normal and its square.

Emails arrive one at a time in an inbox. Let Tn be the time at which the nth email arrives (measured on a continuous scale from some starting point in time). Suppose that the waiting times between emails are i.i.d. Expo(), i.e., T1, T2T1, T3T2,... are i.i.d. Expo(). Each email is non-spam with probability p, and spam with probability q = 1 p (independently of the other emails and of the waiting times). Let X be the time at which the first non-spam email arrives (so X is a continuous r.v., with X = T1 if the 1st email is non-spam, X = T2 if the 1st email is spam but the 2nd one isnt, etc.). (a) Find the mean and variance of X. (b) Find the MGF of X. What famous distribution does this imply that X has (be sure to state its parameter values)? Hint for both parts: Let N be the number of emails until the first non-spam (including that one), and write X as a sum of N terms; then condition on N.

Customers arrive at a store according to a Poisson process of rate customers per hour. Each makes a purchase with probability p, independently. Given that a customer makes a purchase, the amount spent has mean (in dollars) and variance 2. (a) Find the mean and variance of how much a random customer spends (note that the customer may spend nothing). (b) Find the mean and variance of the revenue the store obtains in an 8-hour time interval, using (a) and results from this chapter. (c) Find the mean and variance of the revenue the store obtains in an 8-hour time interval, using the chicken-egg story and results from this chapter.

Freds beloved computer will last an Expo() amount of time until it has a malfunction. When that happens, Fred will try to get it fixed. With probability p, he will be able to get it fixed. If he is able to get it fixed, the computer is good as new again and will last an additional, independent Expo() amount of time until the next malfunction (when again he is able to get it fixed with probability p, and so on). If after any malfunction Fred is unable to get it fixed, he will buy a new computer. Find the expected amount of time until Fred buys a new computer. (Assume that the time spent on computer diagnosis, repair, and shopping is negligible.)

Judit plays in a total of N Geom(s) chess tournaments in her career. Suppose that in each tournament she has probability p of winning the tournament, independently. Let T be the number of tournaments she wins in her career. (a) Find the mean and variance of T. (b) Find the MGF of T. What is the name of this distribution (with its parameters)?

Let X1,...,Xn be i.i.d. r.v.s with mean and variance 2, and n 2. A bootstrap sample of X1,...,Xn is a sample of n r.v.s X 1 ,...,X n formed from the Xj by sampling with replacement with equal probabilities. Let X denote the sample mean of the bootstrap sample: X = 1 n (X 1 + + X n). (a) Calculate E(X j ) and Var(X j ) for each j. (b) Calculate E(X |X1,...,Xn) and Var(X |X1,...,Xn). Hint: Conditional on X1,...,Xn, the X j are independent, with a PMF that puts probability 1/n at each of the points X1,...,Xn. As a check, your answers should be random variables that are functions of X1,...,Xn. (c) Calculate E(X ) and Var(X ). (d) Explain intuitively why Var(X) < Var(X ).

An insurance company covers disasters in two neighboring regions, R1 and R2. Let I1 and I2 be the indicator r.v.s for whether R1 and R2 are hit by the insured disaster, respectively. The indicators I1 may be I2 are dependent. Let pj = E(Ij ) for j = 1, 2, and p12 = E(I1I2). The company reimburses a total cost of C = I1 T1 + I2 T2 to these regions, where Tj has mean j and variance 2 j . Assume that T1 and T2 are independent of each other and independent of I1, I2. (a) Find E(C). (b) Find Var(C).

A certain stock has low volatility on some days and high volatility on other days. Suppose that the probability of a low volatility day is p and of a high volatility day is q = 1 p, and that on low volatility days the percent change in the stock price is N (0, 2 1), while on high volatility days the percent change is N (0, 2 2), with 1 < 2. Let X be the percent change of the stock on a certain day. The distribution is said to be a mixture of two Normal distributions, and a convenient way to represent X is as X = I1X1+I2X2 where I1 is the indicator r.v. of having a low volatility day, I2 = 1I1, Xj N (0, 2 j ), and I1, X1, X2 are independent. (a) Find Var(X) in two ways: using Eves law, and by calculating Cov(I1X1 + I2X2, I1X1 + I2X2) directly. (b) Recall from Chapter 6 that the kurtosis of an r.v. Y with mean and standard deviation is defined by Kurt(Y ) = E(Y ) 4 4 3. Find the kurtosis of X (in terms of p, q, 2 1, 2 2, fully simplified). The result will show that even though the kurtosis of any Normal distribution is 0, the kurtosis of X is positive and in fact can be very large depending on the parameter values.

Show that for any r.v.s X and Y , E(Y |E(Y |X)) = E(Y |X). This has a nice intuitive interpretation if we think of E(Y |X) as the prediction we would make for Y based on X: given the prediction we would use for predicting Y from X, we no longer need to know X to predict Y we can just use the prediction we have! For example, letting E(Y |X) = g(X), if we observe g(X) = 7, then we may or may not know what X is (since g may not be one-to-one). But even without knowing X, we know that the prediction for Y based on X is 7. Hint: Use Adams law with extra conditioning.

A researcher wishes to know whether a new treatment for the disease conditionitis is more eective than the standard treatment. It is unfortunately not feasible to do a randomized experiment, but the researcher does have the medical records of patients who received the new treatment and those who received the standard treatment. She is worried, though, that doctors tend to give the new treatment to younger, healthier patients. If this is the case, then naively comparing the outcomes of patients in the two groups would be like comparing apples and oranges. Suppose each patient has background variables X, which might be age, height and weight, and measurements relating to previous health status. Let Z be the indicator of receiving the new treatment. The researcher fears that Z is dependent on X, i.e., that the distribution of X given Z = 1 is dierent from the distribution of X given Z = 0. In order to compare apples to apples, the researcher wants to match every patient who received the new treatment to a patient with similar background variables who received the standard treatment. But X could be a high-dimensional random vector, which often makes it very dicult to find a match with a similar value of X. The propensity score reduces the possibly high-dimensional vector of background variables down to a single number (then it is much easier to match someone to a person with a similar propensity score than to match someone to a person with a similar value of X). The propensity score of a person with background characteristics X is defined as S = E(Z|X). By the fundamental bridge, a persons propensity score is their probability of receiving the treatment, given their background characteristics. Show that conditional on S, the treatment indicator Z is independent of the background variables X. Hint: It helps to first solve the previous problem. Then show that P(Z = 1|S, X) = P(Z = 1|S). By the fundamental bridge, this is equivalent to showing E(Z|S, X) = E(Z|S).

A group of n friends often go out for dinner together. At their dinners, they play credit card roulette to decide who pays the bill. This means that at each dinner, one person is chosen uniformly at random to pay the entire bill (independently of what happens at the other dinners). (a) Find the probability that in k dinners, no one will have to pay the bill more than once (do not simplify for the case k n, but do simplify fully for the case k>n). (b) Find the expected number of dinners it takes in order for everyone to have paid at least once (you can leave your answer as a finite sum of simple-looking terms). (c) Alice and Bob are two of the friends. Find the covariance between how many times Alice pays and how many times Bob pays in k dinners (simplify fully)

As in the previous problem, a group of n friends play credit card roulette at their dinners. In this problem, let the number of dinners be a Pois() r.v. (a) Alice is one of the friends. Find the correlation between how many dinners Alice pays for and how many free dinners Alice gets (simplify fully). (b) The costs of the dinners are i.i.d. Gamma(a, b) r.v.s, independent of the number of dinners. Find the mean and variance of the total cost (simplify fully).

Paul and n other runners compete in a marathon. Their times are independent continuous r.v.s with CDF F. (a) For j = 1, 2,...,n, let Aj be the event that anonymous runner j completes the race faster than Paul. Explain whether the events Aj are independent, and whether they are conditionally independent given Pauls time to finish the race. (b) For the rest of this problem, let N be the number of runners who finish faster than Paul. Find E(N). (c) Find the conditional distribution of N, given that Pauls time to finish the marathon is t. (d) Find Var(N). Hint: (1) Let T be Pauls time; condition on T and use Eves law. (2) Or use indicator r.v.s.

An actuary wishes to estimate various quantities related to the number of insurance claims and the dollar amounts of those claims for someone named Fred. Suppose that Fred will make N claims next year, where N| Pois(). But is unknown, so the actuary, taking a Bayesian approach, gives a prior distribution based on past experience. Specifically, the prior is Expo(1). The dollar amount of a claim is Log-Normal with parameters and 2 (here and 2 are the mean and variance of the underlying Normal), with and 2 known. The dollar amounts of the claims are i.i.d. and independent of N. (a) Find E(N) and Var(N) using properties of conditional expectation (your answers should not depend on , since is unknown and being treated as an r.v.!). (b) Find the mean and variance of the total dollar amount of all the claims. (c) Find the distribution of N. If it is a named distribution we have studied, give its name and parameters. (d) Find the posterior distribution of , given that it is observed that Fred makes N = n claims next year. If it is a named distribution we have studied, give its name and parameters.

Empirically, it is known that 49% of children born in the U.S. are girls (and 51% are boys). Let N be the number of children who will be born in the U.S. in March of next year, and assume that N is a Pois() random variable, where is known. Assume that births are independent (e.g., dont worry about identical twins). Let X be the number of girls who will be born in the U.S. in March of next year, and let Y be the number of boys who will be born then. (a) Find the joint distribution of X and Y . (Give the joint PMF.) (b) Find E(N|X) and ind E(N|

Let X1, X2, X3 be independent with Xi Expo(i) (so with possibly dierent rates). Recall from Chapter 7 that P(X1 < (a) Find E(X1 + X2 + X3|X1 > 1, X2 > 2, X3 > 3) in terms of 1, 2, 3. (b) Find P (X1 = min(X1, X2, X3)), the probability that the first of the three Exponentials is the smallest. Hint: Restate this in terms of X1 and min(X2, X3). (c) For the case 1 = 2 = 3 = 1, find the PDF of max(X1, X2, X3). Is this one of the important distributions we have studied

A task is randomly assigned to one of two people (with probability 1/2 for each person). If assigned to the first person, the task takes an Expo(1) length of time to complete (measured in hours), while if assigned to the second person it takes an Expo(2) length of time to complete (independent of how long the first person would have taken). Let T be the time taken to complete the task. (a) Find the mean and variance of T. (b) Suppose instead that the task is assigned to both people, and let X be the time taken to complete it (by whoever completes it first, with the two people working independently). It is observed that after 24 hours, the task has not yet been completed. Conditional on this information, what is the expected value of X?

Suppose for this problem that true IQ is a meaningful concept rather than a reified social construct. Suppose that in the U.S. population, the distribution of true IQs is Normal with mean 100 and SD 15. A person is chosen at random from this population to take an IQ test. The test is a noisy measure of true ability: its correct on average but has a Normal measurement error with SD 5. Let be the persons true IQ, viewed as a random variable, and let Y be her score on the IQ test. Then we have Y | N (, 52 ) N (100, 152 ). (a) Find the unconditional mean and variance of Y . (b) Find the marginal distribution of Y . One way is via the MGF. (c) Find Cov(,

A certain genetic characteristic is of interest. It can be measured numerically. Let X1 and X2 be the values of the genetic characteristic for two twin boys. If they are identical twins, then X1 = X2 and X1 has mean 0 and variance 2; if they are fraternal twins, then X1 and X2 have mean 0, variance 2, and correlation . The probability that the twins are identical is 1/2. Find Cov(X1, X2) in terms of p.

The Mass Cash lottery randomly chooses 5 of the numbers from 1, 2,..., 35 each day (without repetitions within the choice of 5 numbers). Suppose that we want to know how long it will take until all numbers have been chosen. Let aj be the average number of additional days needed if we are missing j numbers (so a0 = 0 and a35 is the average number of days needed to collect all 35 numbers). Find a recursive formula for the aj.

Two chess players, Vishy and Magnus, play a series of games. Given p, the game results are i.i.d. with probability p of Vishy winning, and probability q = 1 p of Magnus winning (assume that each game ends in a win for one of the two players). But p is unknown, so we will treat it as an r.v. To reflect our uncertainty about p, we use the prior p Beta(a, b), where a and b are known positive integers and a 2. (a) Find the expected number of games needed in order for Vishy to win a game (including the win). Simplify fully; your final answer should not use factorials or . (b) Explain in terms of independence vs. conditional independence the direction of the inequality between the answer to (a) and 1 + E(G) for G Geom( a a+b ). (c) Find the conditional distribution of p given that Vishy wins exactly 7 out of the first 10 games.

Laplaces law of succession says that if X1, X2,...,Xn+1 are conditionally independent Bern(p) r.v.s given p, but p is given a Unif(0, 1) prior to reflect ignorance about its value, then P(Xn+1 = 1|X1 + + Xn = k) = k + 1 n + 2 . As an example, Laplace discussed the problem of predicting whether the sun will rise tomorrow, given that the sun did rise every time for all n days of recorded history; the above formula then gives (n + 1)/(n + 2) as the probability of the sun rising tomorrow (of course, assuming independent trials with p unchanging over time may be a very unreasonable model for the sunrise problem). (a) Find the posterior distribution of p given X1 = x1, X2 = x2,...,Xn = xn, and show that it only depends on the sum of the xj (so we only need the one-dimensional quantity x1 + x2 + + xn to obtain the posterior distribution, rather than needing all n data points). (b) Prove Laplaces law of succession, using a form of LOTP to find P(Xn+1 = 1|X1 + +Xn = k) by conditioning on p. (The next exercise, which is closely related, involves an equivalent Adams law proof.)

Two basketball teams, A and B, play an n game match. Let Xj be the indicator of team A winning the jth game. Given p, the r.v.s X1,...,Xn are i.i.d. with Xj |p Bern(p). But p is unknown, so we will treat it as an r.v. Let the prior distribution be p Unif(0, 1), and let X be the number of wins for team A. (a) Find E(X) and Var(X). (b) Use Adams law to find the probability that team A will win game j + 1, given that they win exactly a of the first j games. (The previous exercise, which is closely related, involves an equivalent LOTP proof.) Hint: letting C be the event that team A wins exactly a of the first j games, P(Xj+1 = 1|C) = E(Xj+1|C) = E(E(Xj+1|C, p)|C) = E(p|C). (c) Find the PMF of X. (There are various ways to do this, including a very fast way to see it based on results from earlier chapters.)

An election is being held. There are two candidates, A and B, and there are n voters. The probability of voting for Candidate A varies by city. There are m cities, labeled 1, 2,...,m. The jth city has nj voters, so n1 +n2 ++nm = n. Let Xj be the number of people in the jth city who vote for Candidate A, with Xj |pj Bin(nj , pj ). To reflect our uncertainty about the probability of voting in each city, we treat p1,...,pm as r.v.s, with prior distribution asserting that they are i.i.d. Unif(0, 1). Assume that X1,...,Xm are independent, both unconditionally and conditional on p1,...,pm. (a) Find the marginal distribution of X1 and the posterior distribution of p1|X1 = k1. (b) Find E(X) and Var(X) in terms of n and s, where s = n2 1 + n2 2 + + n2 m.