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probability and stochastic modeling

Fundamentals Of Probability With Stochastic Processes 3rd Edition Saeed Ghahramani - Solutions

3. In the experiment of flipping a coin three times, what does the event E = {THH, HTH, HHT, HHH}represent?
2. Last month, an insurance company sold 57 life insurance policies. Define a sample space for the number of claims that the company will receive from the beneficiaries of this group within the next 15 years. What is the event that the company receives at least 3 but no more than 8 claims?
1. From the letters of the wordMISSISSIPPI, a letter is chosen at random.What is a sample space for this experiment?What is the event that the outcome is a vowel?
20. Let X and Y be two given random variables. Prove that Var(X|Y ) = E[X2 |Y ] − E(X|Y )2 .
19. 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).
18. 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
17. 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
16. (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
15. 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
14. 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?
13. 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
12. 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
11. A fair coin is tossed successively. Let Kn be the number of tosses until n consecutive heads occur.(a) Argue that E(Kn | Kn−1 = i) = (i + 1) 1 2 + 4 i + 1 + E(Kn) 51 2 . (b) Show that E(Kn | Kn−1) = Kn−1 + 1 + 1 2 E(Kn). (c) By finding the expected values of both sides of (b) find a
10. 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
9. Prove that, for a Poisson random variable N, if the parameter λ is not fixed and is itself an exponential random variable with parameter 1, then P (N = i) = *1 2 ,i+1 .
8. 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 ).
7. 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
6. 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
5. 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
4. 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).
3. 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
2. 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 ofa, and every ton that is not sold is destroyed at a loss of a/3. How many tons of grain should the farmer
1. 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.
7. Show that if the joint probability density function of X and Y is f (x, y) =    1 2 sin(x + y) if 0 ≤ x ≤ π 2 , 0 ≤ y ≤ π 2 0 elsewhere, then there exists no linear relation between X and Y .
6. Prove that if Cov(X, Y ) = 0, then ρ(X + Y, X − Y ) = Var(X) − Var(Y ) Var(X) + Var(Y ). B
5. Is it possible that for some random variables X and Y , ρ(X, Y ) = 3, σX = 2, and σY = 3?
4. For real numbers α and β, let sgn(αβ) =    1 if αβ > 0 0 if αβ = 0 −1 if αβ < 0. Prove that for random variables X and Y , ρ(α1X + α2, β1Y + β2) = ρ(X, Y ) sgn(α1β1).
3. A stick of length 1 is broken into two pieces at a random point. Find the correlation coefficient and the covariance of these pieces.
2. Let the joint probability density function of X and Y be given by f (x, y) = B sin x sin y if 0 ≤ x ≤ π/2, 0 ≤ y ≤ π/2 0 otherwise. Calculate the correlation coefficient of X and Y .
1. Let X and Y be jointly distributed, with ρ(X, Y ) = 1/2, σX = 2, σY = 3. Find Var(2X − 4Y + 3).
28. Exactly n married couples are living in a small town. What is the variance of the surviving couples after m deaths occur among them? Assume that the deaths occur at random, there are no divorces, and there are no new marriages. Note: This situation involves the Daniel Bernoulli problem
27. Let X be a hypergeometric random variable with probability mass function p(x) = P (X = x) = ; D x
26. Show that if X1, X2, . . . , Xn are random variables and a1, a2, . . . , an are constants, then Var*.n i=1 aiXi , = .n i=1 a2 i Var(Xi) + 2 .. i
25. A fair die is thrown n times. What is the covariance of the number of 1’s and the number of 6’s obtained? Hint: Use the result of Exercise 24.
24. Prove the following generalization of Exercise 23: Cov*.n i=1 aiXi, .m j=1 bjYj , = .n i=1 .m j=1 aibjCov(Xi, Yj ).
23. Show that for random variables X, Y , Z, and W and constantsa, b,c, andd, Cov(aX + bY, cZ + dW ) = ac Cov(X, Z) + bc Cov(Y, Z) + ad Cov(X, W ) + bd Cov(Y, W ). Hint: For a simpler proof, use the results of Exercise 6.
22. Let S be the sample space of an experiment. Let A and B be two events of S. Let IA and IB be the indicator variables for A and B. That is, IA(ω) = B 1 if ω ∈ A 0 if ω ,∈ A, IB(ω) = B 1 if ω ∈ B 0 if ω ,∈ B. Show that IA and IB are positively correlated if and only if P (A | B) > P
21. Let X be a random variable. Prove that Var(X) = min t E 4 (X − t)2 5 . Hint: Let µ = E(X) and look at the expansion of E 4 (X − t)25 = E 4 (X − µ + µ − t)25 . B
20. Let X and Y be jointly distributed with joint probability density function f (x, y) =    1 2 x3 e−xy−x if x > 0, y > 0 0 otherwise. Determine if X and Y are positively correlated, negatively correlated, or uncorrelated. Hint: Note that for all a > 0, # ∞ 0 xne−ax dx =
19. Find the variance of a sum of n randomly and independently selected points from the interval (0, 1).
18. Let X and Y have the following joint probability density function f (x, y) = B 8xy if 0 < x ≤ y < 1 0 otherwise. (a) Calculate Var(X + Y ).(b) Show thatX and Y are not independent. Explain why this does not contradict Exercise 23 of Section 8.2.
17. (Investment) Mr. Kowalski has invested $50,000 in three uncorrelated financial assets: 25% in the first financial asset, 40% in the second one, and 35% in the third one. The annual rates of return for these assets, respectively, are normal random variables with means 12%, 15%, and 18%, and
16. (Investment) Mr. Ingham has invested money in three assets; 18% in the first asset, 40% in the second one, and 42% in the third one. Let r1, r2, and r3 be the annual rate of returns for these three investments, respectively. For 1 ≤ i, j ≤ 3, Cov(ri, rj ) is the ith element in the j th row
15. A voltmeter is used to measure the voltage of voltage sources, such as batteries. Every time this device is used, a random error is made, independent of other measurements, with mean 0 and standard deviation σ. Suppose that we want to measure the voltages, V1 and V2, of two batteries. If a
14. Let X and Y be independent random variables with expected values µ1 and µ2, and variances σ2 1 and σ2 2 , respectively. Show that Var(XY ) = σ2 1 σ2 2 + µ2 1σ2 2 + µ2 2σ2 1 .
13. Mr. Jones has two jobs. Next year, he will get a salary raise of X thousand dollars from one employer and a salary raise of Y thousand dollars from his second. Suppose that X and Y are independent random variables with probability density functions f and g, respectively, where f (x) =  
12. Let X and Y be the coordinates of a random point selected uniformly from the unit disk $ (x, y): x2 + y2 ≤ 1 % . Are X and Y independent? Are they uncorrelated? Why or why not?
11. Prove that if 5 is a random number from the interval [0, 2π], then the dependent random variables X = sin 5 and Y = cos 5 are uncorrelated.
10. Let X and Y be two independent random variables. (a) Show that X−Y and X+Y are uncorrelated if and only if Var(X) =Var(Y ). (b) Show that Cov(X, XY ) = E(Y )Var(X).
9. Prove that Var(X − Y ) = Var(X) + Var(Y ) − 2 Cov(X, Y ).
8. For random variables X and Y , show that Cov(X + Y, X − Y ) = Var(X) − Var(Y ).
7. Show that if X and Y are independent random variables, then for all random variables Z, Cov(X, Y + Z) = Cov(X, Z).
6. For random variables X, Y , and Z, prove that (a) Cov(X + Y, Z) = Cov(X, Z)+Cov(Y, Z). (b) Cov(X, Y + Z) = Cov(X, Y )+Cov(X, Z).
5. In n independent Bernoulli trials, each with probability of success p, let X be the number of successes and Y the number of failures. Calculate E(XY ) and Cov(X, Y ).
4. Thieves stole four animals at random from a farm that had seven sheep, eight goats, and five burros. Calculate the covariance of the number of sheep and goats stolen.
3. Roll a balanced die and let the outcome be X. Then toss a fair coin X times and let Y denote the number of tails. Find Cov(X, Y ) and interpret the result. Hint: Let p(x, y) be the joint probability mass function of X and Y . To save time, use the table for p(x, y) constructed in Example 8.2.
2. Let the joint probability mass function of random variables X and Y be given by p(x, y) =    1 70 x(x + y) if x = 1, 2, 3, y = 3, 4 0 elsewhere. Find Cov(X, Y ).
1. Ann cuts an ordinary deck of 52 cards and displays the exposed card. After Ann places her stack back on the deck, Andy cuts the same deck and displays the exposed card. Counting jack, queen, and king as 11, 12, and 13, let X and Y be the numbers on the cards that Ann and Andy expose,
20. Under what condition does Cauchy-Schwarz’s inequality become equality?
19. From an urn that contains a large number of red and blue chips, mixed in equal proportions, 10 chips are removed one by one and at random. The chips that are removed before the first red chip are returned to the urn. The first red chip, together with all those that follow, is placed in another
18. Let {X1, X2, . . . , Xn} be a sequence of continuous, independent, and identically distributed random variables. Let N = min{n: X1 ≥ X2 ≥ X3 ≥ ··· ≥ Xn−1, Xn−1 < Xn}. Find E(N ).
17. Let X and Y be nonnegative random variables with an arbitrary joint probability distribution function. Let I (x, y) = B 1 if X > x, Y > y 0 otherwise. (a) Show that E ∞ 0 E ∞ 0 I (x, y) dx dy = XY. (b) By calculating expected values of both sides of part (a), prove that E(XY ) = E ∞ 0 E
16. (Pattern Appearance) In successive independent flips of a fair coin, what is the expected number of trials until the pattern THTHTTHTHT appears?
15. 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: See Exercise 9, Section 3.2.
14. There are 25 students in a probability class. What is the expected number of the days of the year that are birthdays of at least two students? Assume that the birthrates are constant throughout the year and that each year has 365 days.
13. There are 25 students in a probability class. What is the expected number of birthdays that belong only to one student? Assume that the birthrates are constant throughout the year and that each year has 365 days. Hint: Let Xi = 1 if the birthday of the ith student is not the birthday of any
12. Suppose that 80 balls are placed into 40 boxes at random and independently. What is the expected number of the empty boxes?
11. A coin is tossed n times (n > 4). What is the expected number of exactly three consecutive heads? Hint: Let E1 be the event that the first three outcomes are heads and the fourth outcome is tails. For 2 ≤ i ≤ n − 3, let Ei be the event that the outcome (i − 1) is tails, the outcomes i,
10. Let {X1, X2, . . . , Xn} be a sequence of independent random variables with P (Xj = i) = pi (1 ≤ j ≤ n and i ≥ 1). Let hk = /∞ i=k pi. Using Theorem 10.2, prove that E 4 min(X1, X2, . . . , Xn) 5 = .∞ k=1 hn k .
9. Solve the following problem posed by Michael Khoury, U.S. Mathematics Olympiad Member, in “The Problem Solving Competition,” Oklahoma Publishing Company and the American Society for Communication of Mathematics, February 1999. Bob is teaching a class with n students. There are n desks in the
8. (Pattern Appearance) Suppose that random digits are generated from the set {0, 1, . . . , 9} independently and successively. Find the expected number of digits to be generated until the pattern (a) 007 appears, (b) 156156 appears, (c) 575757 appears
7. A cultural society is arranging a party for its members. The cost of a band to play music, the amount that the caterer will charge, the rent of a hall to give the party, and other expenses (in dollars) are uniform random variables over the intervals (1300, 1800), (1800, 2000), (800, 1200), and
6. An absentminded professor wrote n letters and sealed them in envelopes without writing the addresses on the envelopes. Having forgotten which letter he had put in which envelope, he wrote the n addresses on the envelopes at random. What is the expected number of the letters addressed correctly?
5. A company puts five different types of prizes into their cereal boxes, one in each box and in equal proportions. If a customer decides to collect all five prizes, what is the expected number of the boxes of cereals that he or she should buy?
4. Let the joint probability density function of random variables X and Y be f (x, y) = B 2e−(x+2y) if x ≥ 0, y ≥ 0 0 otherwise. Find E(X), E(Y ), and E(X2 + Y 2).
3. Let X, Y , and Z be three independent random variables such that E(X) = E(Y ) = E(Z) = 0, and Var(X) =Var(Y ) =Var(Z) = 1. Calculate E 4 X2(Y + 5Z)2 5 .
2. A calculator is able to generate random numbers from the interval (0, 1). We need five random numbers from (0, 2/5). Using this calculator, how many independent random numbers should we generate, on average, to find the five numbers needed?
1. Let the probability density function of a random variable X be given by f (x) = B |x − 1| if 0 ≤ x ≤ 2 0 otherwise. Find E(X2 + X).
10. Let X1, X2, and X3 be independent random variables from (0, 1). Find the probability density function and the expected value of the midrange of these random variables [X(1) + X(3)]/2
9. A bar of length $ is broken into three pieces at two random spots. What is the probability that the length of at least one piece is less than $/20?
8. A system consists of n components whose lifetimes form an independent sequence of random variables. Suppose that the system functions as long as at least one of its components functions. Let F1, F2, ... , Fn be the distribution functions of the lifetimes of the components of the system. In terms
7. A system consists of n components whose lifetimes form an independent sequence of random variables. In order for the system to function, all components must function. Let F1, F2, ... , Fn be the distribution functions of the lifetimes of the components of the system. In terms of F1, F2, ... ,
6. Alvie, a marksman, fires seven independent shots at a target. Suppose that the probabilities that he hits the bull’s-eye, he hits the target but not the bull’s-eye, and he misses the target are 0.4, 0.35, and 0.25, respectively. What is the probability that he hits the bull’s-eye three
5. A fair die is tossed 18 times. What is the probability that each face appears three times?
4. The joint probability density function of random variables X, Y , and Z is given by f (x, y, z) = B c(x + y + 2z) if 0 ≤ x, y, z ≤ 1 0 otherwise. (a) Determine the value ofc. (b) Find P (X < 1/3 | Y < 1/2, Z < 1/4).
3. Suppose that n points are selected at random and independently inside the cube # = $ (x, y, z): − a ≤ x ≤a, −a ≤ y ≤a, −a ≤ z ≤ a % . Find the probability that the distance of the nearest point to the center is at least r (r < a).
2. Let X be the smallest number obtained in rolling a balanced die n times. Calculate the probability distribution function and the probability mass function of X.
1. An urn contains 100 chips of which 20 are blue, 30 are red, and 50 are green. Suppose that 20 chips are drawn at random and without replacement. Let B, R, and G be the number of blue, red, and green chips, respectively. Calculate the joint probability mass function of B, R, and G.
9. Customers enter a department store at the rate of three per minute, in accordance with a Poisson process. If 30% of them buy nothing, 20% pay cash, 40% use charge cards, and 10% write personal checks, what is the probability that in five operating minutes of the store, five customers use charge
8. (Genetics) Let p and q be positive numbers with p + q = 1. For a gene with dominant allele A and recessive allelea, let p2, 2pq, and q2 be the probabilities that a randomly selected person from a population has genotype AA, Aa, and aa, respectively. A group of six members of the population is
7. (Genetics) As we know, in humans, for blood type, there are three alleles A, B, and O. The alleles A and B are codominant to each other and dominant to O. A man of genotype AB marries a woman of genotype BO. If they have six children, what is the probability that three will have type B blood,
6. Suppose that the ages of 30% of the teachers of a country are over 50, 20% are between 40 and 50, and 50% are below 40. In a random committee of 10 teachers from this country, two are above 50. What is the probability mass function of those who are below 40?
5. Suppose that 50% of the watermelons grown on a farm are classified as large, 30% as medium, and 20% as small. Joanna buys five watermelons at random from this farm. What is the probability that (a) at least two of them are large; (b) two of them are large, two are medium, and one is small; (c)
4. At a certain college, 16% of the calculus students get A’s, 34% B’s, 34% C’s, 14% D’s, and 2% F’s. What is the probability that, of 15 calculus students selected at random, five get B’s, five C’s, two D’s, and at least two A’s?
3. Suppose that each day the price of a stock moves up 1/8 of a point with probability 1/4, remains the same with probability 1/3, and moves down 1/8 of a point with probability 5/12. If the price fluctuations from one day to another are independent, what is the probability that after six days the
2. An urn contains 100 chips of which 20 are blue, 30 are red, and 50 are green. We draw 20 chips at random and with replacement. Let B, R, and G be the number of blue, red, and green chips, respectively. Calculate the joint probability mass function of B, R, and G.
hours?
1. Light bulbs manufactured by a certain factory last a random time between 400 and 1200 hours. What is the probability that, of eight such bulbs, three burn out before 550 hours, two burn out after 800 hours, and three burn out after 550 but before 800
10. Let X1, X2, . . . , Xn be n independently randomly selected points from the interval (0, θ), θ > 0. Prove that E(R) = n − 1 n + 1 θ, where R = X(n) − X(1) is the range of these points. Hint: Use part (a) of Exercise 9. Also compare this with Exercise 18, Section 9.1.
9. Let X1, X2, . . . , Xn be a random sample of size n from a population with continuous probability distribution function F and probability density function f . (a) Calculate the probability density function of the sample range, R = X(n) − X(1). (b) Use (a) to find the probability density

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