In a box, Lynn has b batteries of which d are dead. She

A fair coin is tossed until two tails occur successively. Find the expected number of the tosses required. Hint: Let X = B 1 if the first toss results in tails 0 if the first toss results in heads, and condition on X.

The orders received for grain by a farmer add up to X tons, where X is a continuous random variable uniformly distributed over the interval (4, 7). Every ton of grain sold brings a profit of a, and every ton that is not sold is destroyed at a loss of a/3. How many tons of grain should the farmer produce to maximize his expected profit? Hint: Let Y (t) be the profit if the farmer produces t tons of grain. Then E 4 Y (t)5 = E 8 aX a 3 (t X)9 P (X < t) + E(at)P (X t).

In a box, Lynn has b batteries of which d are dead. She tests them randomly and one by one. Every time that a good battery is drawn, she will return it to the box; every time that a dead battery is drawn, she will replace it by a good one. (a) Determine the expected value of the number of good batteries in the box after n of them are checked. (b) Determine the probability that on the nth draw Lynn draws a good battery. Hint: Let Xn be the number of good batteries in the box after n of them are checked. Show that E(Xn | Xn1) = 1 + * 1 1 b , Xn1. Then, by computing the expected value of this random variable, find a recursive relation between E(Xn) and E(Xn1). Use this relation and induction to prove that E(Xn) = b d * 1 1 b ,n . Note that n should approach to get E(Xn) = b. For part (b), let En be the event that on the nth draw she gets a good battery. By conditioning on Xn1 prove that P (En) = E(Xn1)/b.

For given random variables Y and Z, let X = B Y with probability p Z with probability 1 p. Find E(X) in terms of E(Y ) and E(Z).

A typist, on average, makes three typing errors in every two pages. If pages with more than two errors must be retyped, on average how many pages must she type to prepare a report of 200 pages? Assume that the number of errors in a page is a Poisson random variable. Note that some of the retyped pages should be retyped, and so on. Hint: Find p, the probability that a page should be retyped. Let Xn be the number of pages that should be typed at least n times. Show that E(X1) = 200p, E(X2) = 200p2, ... , E(Xn) = 200pn. The desired quantity is E !/ i=1 Xi " , which can be calculated using relation (10.2).

In data communication, usually messages sent are combinations of characters, and each character consists of a number of bits. A bit is the smallest unit of information and is either 1 or 0. Suppose that the length of a character (in bits) is a geometric random variable with parameter p. Suppose that a message is combined of K characters, where K is a random variable with mean and variance 2. If the lengths of characters of a message are independent of each other and of K, and if it takes a sender 1000 bits per second to emit a message, find the expected value of T , the time it will take the sender to emit a message.

From an ordinary deck of 52 cards, cards are drawn at random, one by one and without replacement until a heart is drawn. What is the expected value of the number of cards drawn? Hint: Consider a deck of cards with 13 hearts and 39 n nonheart cards. Let Xn be the number of cards to be drawn before the first heart is drawn. Let Y = B 1 if the first card drawn is a heart 0 otherwise. By conditioning on Y , find a recursive relation between E(Xn) and E(Xn+1). Use E(X39) = 0 to show that E(Xi) = (39 i)/14. The answer is 1 + E(X0). (For a totally different solution see Exercise 15, Section 10.1.)

Suppose that X and Y are independent random variables with probability density functions f and g, respectively. Use conditioning technique to calculate P (X < Y ).

Prove that, for a Poisson random variable N, if the parameter is not fixed and is itself an exponential random variable with parameter 1, then P (N = i) = *1 2 ,i+1 .

Suppose that X and Y represent the amount of money in the wallets of players A and B, respectively. Let X and Y be jointly uniformly distributed on the unit square [0, 1][0, 1]. A and B each places his wallet on the table. Whoever has the smallest amount of money in his wallet, wins all the money in the other wallet. Let WA be the amount of money that player A will win. Show that E(WA) = 0. (For a history of this problem, see the Wallet Paradox in Exercise 22, Review Problems, Chapter 8.)

A fair coin is tossed successively. Let Kn be the number of tosses until n consecutive heads occur. (a) Argue that E(Kn | Kn1 = i) = (i + 1) 1 2 + 4 i + 1 + E(Kn) 51 2 . (b) Show that E(Kn | Kn1) = Kn1 + 1 + 1 2 E(Kn). (c) By finding the expected values of both sides of (b) find a recursive relation between E(Kn) and E(Kn1). (d) Note that E(K1) = 2. Use this and (c) to find E(Kn).

In Rome, tourists arrive at a historical monument according to a Poisson process, on average, one every five minutes. There are guided tours that depart (a) whenever there is a group of 10 tourists waiting to take the tour, or (b) one hour has elapsed from the time the previous tour began. It is the policy of the Tourism Department that a tour will only run for less than 10 people if the last guided tour left one hour ago. If in any one hour period, there will always be tourists arriving to take the tour, find the expected value of the time between two consecutive tours.

During an academic year, the admissions office of a small college receives student applications at a Poisson rate of 5 per day. It is a policy of this college to double its student recruitment efforts if no applications arrive for two consecutive business days. Find the expected number of business days until a time when the college needs to double its recruitment efforts. Do the admission officers need to worry about this policy at all? Hint: Let X1 be the time until the first application arrives. Let X2 be the time between the first and second applications, and so forth. Let N be the first integer for which X1 2, X2 2, . . . , XN 2, XN+1 > 2. The time that the admissions office has to wait before doubling its student recruitment efforts is SN+1 = X1 + X2 + + XN+1. Find SN+1 by conditioning on N.

Each time that Steven calls his friend Adam, the probability that Adam is available to talk with him is p independently of other calls. On average, after how many calls has Steven not missed Adam k consecutive times?

Recently, Larry taught his daughter Emily how to play backgammon. To encourage Emily to practice this game, Larry decides to play with her until she wins two of the recent three games. If the probability that Emily wins a game is 0.35 independently of all preceding and future games, find the expected number of games to be played.

(Genetics) Hemophilia is a sex-linked disease with normal allele H dominant to the mutant allele h. Kim and John are married, and John is phenotypically normal. Suppose that, in the entire population, the frequencies of H and h are 0.98 and 0.02, respectively. If Kim and John have four sons and three daughters, what is the expected number of their hemophilic children?

A spice company distributes cinnamon in one-pound bags. Suppose that the Food and Drug Administration (FDA) considers more than 500 insect fragments in one bag excessive and hence unacceptable. To meet the standards of the FDA, the quality control division of the company begins inspecting the bags of the cinnamon at time t = 0 according to the following scheme. It inspects each bag with probability until it encounters an unacceptable bag. At that point, the division inspects every single bag until it finds m consecutive acceptable bags. When this happens, the division has completed one inspection cycle. It then resumes its normal inspection process. Inspected bags that are found with excessive numbers of insect fragments are sent back for further cleaning. Let p be the probability that a bag is acceptable, independent of the number of insect fragments in other bags. Find the expected value of the number of bags inspected in one inspection cycle.

Suppose that a device is powered by a battery. Since an uninterrupted supply of power is needed, the device has a spare battery. When the battery fails, the circuit is altered electronically to connect the spare battery and remove the failed battery from the circuit. The spare battery then becomes the working battery and emits a signal to alert the user to replace the failed battery. When that battery is replaced, it becomes the new spare battery. Suppose that the lifetimes of the batteries used are independent uniform random variables over the interval (0, 1), where the unit of measurement is 1000 hours. For 0 < t 1, on average, how many batteries are changed by time t? How many are changed, on average, after 950 hours of operation?

Let X and Y be continuous random variables. Prove that E 4!X E(X|Y )"25 = E(X2 ) E 4 E(X|Y )25 . Hint: Let Z = E(X|Y ). By conditioning on Y and using Example 10.23, first show that E(XZ) = E(Z2).

Let X and Y be two given random variables. Prove that Var(X|Y ) = E[X2 |Y ] E(X|Y )2 .

Prove Theorem 10.8.