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Fundamentals Of Probability With Stochastic Processes 4th Edition Saeed Ghahramani - Solutions
Professors Davidson and Johnson from the University of Victoria in Canada gave the following problem to their students in a finite math course:An urn contains N balls of which B are black and N −B are white; n balls are chosen at random and without replacement from the urn. If X is the number of
Every month, a large lot of 2000 fuses manufactured by a certain company is shipped to a major retail store. The policy of the store is to return a lot if more than 2% of its fuses are defective. Since 2% of 50 fuses is 1 fuse, the store selects 50 of the 2000 fuses at random and tests them all. If
In a community of a + b potential voters, a are pro-choice and b (b < a)are pro-life. Suppose that a vote is taken to determine the will of the majority with regard to legalizing abortion. If n (n
In 500 independent calculations a scientist has made 25 errors. If a second scientist checks seven of these calculations randomly, what is the probability that he detects two errors? Assume that the second scientist will definitely find the error of a false calculation
A smoking mathematician carries two matchboxes, one in his right pocket and one in his left pocket. Whenever he wants to smoke, he selects a pocket at random and takes a match from the box in that pocket. If each matchbox initially contains N matches, what is the probability that when the
Two gamblers play a game in which in each play gambler A beats B with probability p, 0 < p < 1, and loses to B with probability q = 1−p. Suppose that each play results in a forfeiture of $1 for the loser and in no change for the winner. If player A initially has a dollars and player B has b
Sharon and Ann play a series of backgammon games until one of them wins five games. Suppose that the games are independent and the probability that Sharon wins a game is 0.58.(a) Find the probability that the series ends in seven games.(b) If the series ends in seven games, what is the probability
A father asks his sons to cut their backyard lawn. Since he does not specify which of the three sons is to do the job, each boy tosses a coin to determine the odd person, who must then cut the lawn. In the case that all three get heads or tails, they continue tossing until they reach a decision.
From an ordinary deck of 52 cards we draw cards at random, with replacement, and successively until an ace is drawn. What is the probability that at least 10 draws are needed?
34. Let X be a Poisson random variable with parameter λ. Show that the maximum of P(X = i) occurs at [λ], where [λ] is the greatest integer less than or equal to λ.Hint: Let p be the probability mass function of X. Prove thatUse this to find the values of i at which p is increasing and the
33. In a forest, the number of trees that grow in a region of area R has a Poisson distribution with mean λR, where λ is a given positive number.(a) Find the probability that the distance from a certain tree to the nearest tree is more than d.(b) Find the probability that the distance from a
31. LetN(t), t ≥ 0be a Poisson process with rate λ. Suppose that N(t) is the total number of two types of events that have occurred in [0, t]. Let N1(t) and N2(t) be the total number of events of type 1 and events of type 2 that have occurred in [0, t], respectively. If events of type 1 and type
30. LetN(t), t ≥ 0be a Poisson process.What is the probability of (a) an even number of events in (t, t + α); (b) an odd number of events in (t, t + α)?
29. Balls numbered 1,2, . . . , and n are randomly placed into cells numbered 1, 2, . . . , and n. Therefore, for 1 ≤ i ≤ n and 1 ≤ j ≤ n, the probability that ball i is in cell j is 1/n. For each i, 1 ≤ i ≤ n, if ball i is in cell i, we say that a match has occurred at cell i.(a) What
26. Suppose that, on the Richter scale, earthquakes of magnitude 5.5 or higher have probability 0.015 of damaging certain types of bridges. Suppose that such intense earthquakes occur at a Poisson rate of 1.5 per ten years. If a bridge of this type is constructed to last at least 60 years, what is
25. On a certain two-lane north-south highway, there is a T junction. Cars arrive at the junction according to a Poisson process, on the average four per minute. For cars to turn left onto the side street, the highway is widened by the addition of a left-turn lane that is long enough to accommodate
24. Let X be a Poisson random variable with parameter λ. LetFind the probability mass function of Y . 0 if X is zero or even Y If X is odd.
23. A wire manufacturing company has inspectors to examine the wire for fractures as it comes out of a machine. The number of fractures is distributed in accordance with a Poisson process, having one fracture on the average for every 60 meters of wire. One day an inspector has to take an emergency
22. A company is located in a region that is prone to dust storms. The company has purchased an insurance policy that pays $200,000 for each dust storm in a year except the first one. If the number of the dust storms per year is a Poisson random variable with mean 0.83, what is the expected amount
20. Accidents occur at an intersection at a Poisson rate of three per day.What is the probability that during January there are exactly three days (not necessarily consecutive) without any accidents?
18. Suppose that, for a telephone subscriber, the number of wrong numbers is Poisson, at a rate of λ = 1 per week. A certain subscriber has not received any wrong numbers from Sunday through Friday. What is the probability that he receives no wrong numbers on Saturday either?
17. A pharmaceutical company has developed an Ebola vaccine, which is 98.5% of the time effective. If 5000 people have received the vaccine, and sooner or later all of them will be exposed to the Ebola virus, what is the probability that at most 70 of them will become sick with Ebola? Assume that a
15. Suppose that in Japan earthquakes occur at a Poisson rate of three per week.What is the probability that the next earthquake occurs after two weeks?
11. The department of mathematics of a state university has 26 faculty members. For i =0, 1, 2, 3, find pi, the probability that i of them were born on IndependenceDay (a) using the binomial distribution; (b) using the Poisson distribution. Assume that the birth rates are constant throughout the
4. By Example 2.24, the probability that a poker hand is a full house is 0.0014.What is the probability that in 500 random poker hands there are at least two full houses?
Let N(t) be the number of earthquakes that occur at or prior to time t worldwide.Suppose thatN(t) : t ≥ 0is a Poisson process with rate λ and the probability that the magnitude of an earthquake on the Richter scale is 5 or more is p. Find the probability of k earthquakes of such magnitudes at
A fisherman catches fish at a Poisson rate of two per hour from a large lake with lots of fish. Yesterday, he went fishing at 10:00 A.M. and caught just one fish by 10:30 and a total of three by noon.What is the probability that he can duplicate this feat tomorrow?
Suppose that earthquakes occur in a certain region of California, in accordance with a Poisson process, at a rate of seven per year.(a) What is the probability of no earthquakes in one year?(b) What is the probability that in exactly three of the next eight years no earthquakes will occur?
Suppose that children are born at a Poisson rate of five per day in a certain hospital. What is the probability that (a) at least two babies are born during the next six hours;(b) no babies are born during the next two days?
Consider a class of size k of unrelated students. Assuming that the birth rates are constant throughout the year and each year has 365 days, with probability p = 1/365, each pair of the students has the same birthday.Let X be the number of such pairs. Argue that X is approximately a Poisson random
Suppose that n raisins are thoroughly mixed in dough. If we bake k raisin cookies of equal sizes from this mixture, what is the probability that a given cookie contains at least one raisin?
Every week the average number of wrong-number phone calls received by a certain mail-order house is seven. What is the probability that they will receive (a) two wrong calls tomorrow; (b) at least one wrong call tomorrow?
2. Suppose that a prescription drug causes 7 side effects each with probability of 0.05 independently of the others.What is the probability that a patient taking this drug experiences at most 2 side effects.
1. What is the probability that at least two of the six members of a family are not born in the fall? Assume that all seasons have the same probability of containing the birthday of a person selected randomly.
38. While Rose always tells the truth, four of her friends, Albert, Brenda, Charles, and Donna, tell the truth randomly only in one out of three instances, independent of each other. Albert makes a statement. Brenda tells Charles that Albert’s statement is the truth.Charles tells Donna that
37. An urn contains n balls whose colors, red or blue, are equally probable.For example, the probability that all of the balls are red is (1/2)n.If in drawing k balls from the urn, successively with replacement and randomly, no red balls appear, what is the probability that the urn contains no
36. (a) What is the probability of an even number of successes in n independent Bernoulli trials?Hint: Let rn be the probability of an even number of successes in n Bernoulli trials. By conditioning on the first trial and using the law of total probability (Theorem 3.3), show that for n ≥ 1,(b)
35. The post office of a certain small town has only one clerk to wait on customers. The probability that a customer will be served in any given minute is 0.6, regardless of the time that the customer has already taken. The probability of a new customer arriving is 0.45, regardless of the number of
34. How many games of poker occur until a preassigned player is dealt at least one straight flush with probability of at least 3/4? (See Exercise 35 of Section 2.4 for a definition of a straight flush.)
33. In Exercise 32, suppose that a message consisting of six characters is transmitted. If each character consists of seven bits, what is the probability that the message is erroneously received, but none of the errors is detected by the parity check?
32. The simplest error detection scheme used in data communication is parity checking.Usually messages sent consist of characters, each character consisting of a number of bits (a bit is the smallest unit of information and is either 1 or 0). In parity checking, a 1 or 0 is appended to the end of
31. Suppose that an aircraft engine will fail in flight with probability 1−p independently of the plane’s other engines. Also suppose that a plane can complete the journey successfully if at least half of its engines do not fail.(a) Is it true that a four-engine plane is always preferable to a
30. Let X be a binomial random variable with the parameters (n, p). Prove that n E(X)=2 x=1 n p (1 - p)-np - np + np.
29. A game often played in carnivals and gambling houses is called chuck-a-luck, where a player bets on any number 1 through 6. Then three fair dice are tossed. If one, two, or all three land the same number as the player’s, then he or she receives one, two, or three times the original stake plus
28. In a community, a persons are pro-choice, b (b a − b)are undecided. Suppose that there will be a vote to determine the will of the majority with regard to legalizing abortion. If by then all of the undecided persons make up their minds, what is the probability that those who are pro-life
27. Consider the following problem posed by Michael Khoury, U.S. Math Olympiad Team Member, in “The Problem Solving Competition,” Oklahoma Publishing Company and the American Society for the Communication of Mathematics, February 1999.Bob is teaching a class with n students. There are n desks
26. A computer network consists of several stations connected by various media (usually cables). There are certain instances when no message is being transmitted. At such “suitable instances,” each station will send a message with probability p independently of the other stations. However, if
25. A woman and her husband want to have a 95% chance for at least one boy and at least one girl.What is the minimumnumber of children that they should plan to have?Assume that the events that a child is a girl and a boy are equiprobable and independent of the gender of other children born in the
24. Vincent is a patient with the life threatening blood cancer leukemia, and he is in need of a bone marrow transplant. He asks n people whether or not they are willing to donate bone marrow to him if they are a close bone marrow match for him. Suppose that each person’s response, independently
23. Edward’s experience shows that 7% of the parcels he mails will not reach their destination.He has bought two books for $20 each and wants to mail them to his brother. If he sends them in one parcel, the postage is $5.20, but if he sends them in separate parcels, the postage is $3.30 for each
22. A certain rare blood type can be found in only 0.05% of people. If the population of a randomly selected group is 3000, what is the probability that at least two persons in the group have this rare blood type?
21. Let X be the number of sixes obtained when a balanced die is tossed five times. Find the probability mass function of Y = (X − 3)2.
20. What are the expected value and variance of the number of full house hands in n poker hands? A poker hand consists of five randomly selected cards from an ordinary deck of 52 cards. It is a full house if three cards are of one denomination and two cards are of another denomination: for example,
19. A certain basketball player makes a foul shot with probability 0.45. Determine for what value of k the probability of k baskets in 10 shots is maximum, and find this maximum probability.
18. Dr. Willis is teaching two sections of Calculus I, each with 25 students. Suppose that each student will earn a passing grade, independently of other students, with probability of 0.82.What is the probability that in one of Dr. Willis’s classes more than 22 students will earn a passing grade
17. From the set {x: 0 ≤ x ≤ 1}, 100 independent numbers are selected at random and rounded to three decimal places. What is the probability that at least one of them is 0.345?
16. Even though there are some differences between the taste of Max Cola and the taste of Golden Cola, most people cannot tell the difference. Pedro claims that in the absence of brand information, he can say whether a cup of cola is Max or Golden by tasting the drink. A statistician decided to
15. Suppose that each day the price of a stock moves up 1/8th of a point with probability 1/3 and moves down 1/8th of a point with probability 2/3. If the price fluctuations from one day to another are independent, what is the probability that after six days the stock has its original price?
14. On average, how many times should Ernie play poker in order to be dealt a straight flush(royal flush included)? (See Exercise 35 of Section 2.4 for definitions of a royal and a straight flush.)
13. Let X be a binomial random variable with parameters (n, p) and probability mass function p(x). Prove that if (n + 1)p is an integer, then p(x) is maximum at two different points. Find both of these points.
12. From the interval (0, 1), five points are selected at random and independently. What is the probability that (a) at least two of them are less than 1/3; (b) the first decimal point of exactly two of them is 3?
11. If two fair dice are rolled 10 times, what is the probability of at least one 6 (on either die) in exactly five of these 10 rolls?
10. Suppose that the Internal Revenue Service will audit 20%of income tax returns reporting an annual gross income of over $80,000.What is the probability that of 15 such returns, at most four will be audited?
9. Only 60% of certain kinds of seeds germinate when planted under normal conditions.Suppose that four such seeds are planted, and X denotes the number of those that will germinate. Find the probability mass functions of X and Y = 2X + 1.
8. Let X be a discrete random variable with probability mass function p given byFind the probability mass functions of the random variables Y = |X| and Z = X2. X -1 0 1 p(x) 2/9 4/9 3/9 other values 0
7. A manufacturer of nails claims that only 3% of its nails are defective. A random sample of 24 nails is selected, and it is found that two of them are defective. Is it fair to reject the manufacturer’s claim based on this observation?
6. A box contains 30 balls numbered 1 through 30. Suppose that five balls are drawn at random, one at a time, with replacement. What is the probability that the numbers of two of them are prime?
5. In a state where license plates contain six digits, what is the probability that the license number of a randomly selected car has two 9’s? Assume that each digit of the license number is randomly selected from {0, 1, . . . , 9}.
4. LetX be a Bernoulli random variable with parameter p, 0 < p < 1. Find the probability mass function of Y = 1 − X, E(Y ), and Var(Y ).
Two proofreaders, Ruby and Myra, read a book independently and found r and m misprints, respectively. Suppose that the probability that a misprint is noticed by Ruby is p and the probability that it is noticed by Myra is q, where these two probabilities are independent. If the number of misprints
A town of 100,000 inhabitants is exposed to a contagious disease. If the probability that a person becomes infected is 0.04, what is the expected number of people who become infected?
We will now examine an elementary example of a random walk. In Chapter 12, we will revisit this concept and its applications.Suppose that a particle is at 0 on the integer number line and suppose that at step 1, the particle will move to 1 with probability p, 0 < p < 1, and will move to −1 with
Let p be the probability that a randomly chosen person is pro-life, and let X be the number of persons pro-life in a random sample of size n. Suppose that, in a particular random sample of n persons, k are pro-life. Show that P(X = k) is maximum for ˆp = k/n.That is, ˆp is the value of p that
A Canadian pharmaceutical company has developed three types of Ebola virus vaccines. Suppose that vaccines I, II, and III are effective 92%, 88%, and 96% of the time, respectively. Assume that a person reacts to the Ebola vaccine independently of other people.The company sends to Guinea hundreds of
Machuca’s favorite bridge hands are those with at least two aces. Determine the number of times he should play in order to have a chance of 90% or more to get at least two favorite hands? Recall that a bridge hand consists of 13 randomly selected cards from an ordinary deck of 52 cards.
Suppose that jury members decide independently and that each with probability p (0 < p < 1) makes the correct decision. If the decision of the majority is final, which is preferable: a three-person jury or a single juror?
In a small town, out of 12 accidents that occurred in June 1986, four happened on Friday the 13th. Is this a good reason for a superstitious person to argue that Friday the 13th is inauspicious?
In a county hospital 10 babies, of whom six were boys, were born last Thursday.What is the probability that the first six births were all boys? Assume that the events that a child born is a girl or is a boy are equiprobable
A restaurant serves 8 entr´ees of fish, 12 of beef, and 10 of poultry. If customers select from these entr´ees randomly, what is the probability that two of the next four customers order fish entr´ees?
10. Under what conditions on α, β, γ, and θ is the following a distribution function? a t
9. At a department store, summer polo shirts are sold from April 1st until the end of September. Suppose that for each polo shirt the department store sells the net profit is $15.00, and for each shirt unsold by the end of September the net loss is $10.00. Also suppose that, at the department
8. At a university, professors are allowed to check out as many books as they wish from the library. Let X be the number of books checked out by a random professor visiting the library. Suppose that P(X 1) 7/13, P(X = 1 or X = 2)=6/13, P(X > 2) = 3/13, P(X = 3) = P(X > 3). == For i = 0, 1, 2, 3,
7. The season finale of a Belavian reality television series, “Belavia’s Got Talent,” was held in an oval auditorium which has 25 rows of seats. The first row has 10 seats, and each succeeding row has 2 more seats than the previous row. All the seats in the auditorium were occupied, and
6. Let x be a nonnegative real number. By [x], we mean the greatest integer less than or equal to x. For example, [1.2] = 1, [5.8] = 5, and [0.8] = 0. In a box, there are n identical balls numbered from 1 to n. Let X be the number on a ball that is drawn at random from the box. Find F, the
5. To help defray hospitalization expenses due to accidents, the insurance company, Joseph Accident and Health, offers a hospital indemnity plan in which an injured customer is paid a lump sum daily amount of a dollars per day up to three days of hospitalization and a/2 dollars per day thereafter
4. Let X be the number of bagels purchased by a random customer from the Longmeadow Bagel Shop. Let F be the distribution function of X, and suppose that F(0) = 4/33, F(1) = 16/33, F(2) = 26/33, and F(3) = 30/33. If P(X = 4) = P(X > 4), for i = 0, 1, 2, 3, 4, find P(X = i).
3. In the front yard of a doctor’s clinic, there are exactly six parking spaces next to each other in a row. Suppose that at a time when all six parking spaces are occupied, two of the cars leave at random and their spots remain empty. Let X be the number of cars between the empty spots. Find p,
2. A large company has w women and m men in retirement age. If a random set of n, n ≤ w+m, of these employees decides to retire next year, what is the probability mass function of the number of women who will retire next year?
1. From a leap year calendar, a month is selected at random. Let X be the number of days in that month. Find E(X) and σX. If the month is selected from a non-leap year, on average, how many days does that month have?
11. If the motor of a certain commercial grade dishwasher fails within the first year of purchase, the insurance company pays $2500 for its replacement or repair. Thereafter, each year, the company’s payment for motor failure will be $500 less than the previous year until the sixth year. If the
10. From the set of families with three children a family is selected at random, and the number of its boys is denoted by the random variable X. Find the probability mass function and the distribution functions of X. Assume that in a three-child family all gender distributions are equally probable.
9. Experience shows that X, the number of customers entering a post office, during anyperiod of length t, is a random variable the probability mass function of which is of the form(a) Determine the value of k.(b) Compute P(X 1). P(i)=k(2t) i! i i = 0, 1, 2,....
8. The fasting blood-glucose levels of 30 children are as follows.Let X be the fasting blood-glucose level of a child chosen randomly from this group.Find the distribution function of X. 58 62 80 58 64 76 80 80 80 58 62 64 76 76 58 64 62 80 58 58 80 64 58 62 76 62 64 80 62 76
7. Let X be the amount (in fluid ounces) of soft drink in a randomly chosen bottle from company A, and Y be the amount of soft drink in a randomly chosen bottle from company B. A study has shown that the distributions of X and Y are as follows:Find E(X), E(Y ), Var(X), and Var(Y ) and interpret
6. The annual amount of rainfall (in centimeters) in a certain area is a random variable with the distribution functionWhat is the probability that next year it will rain (a) at least 6 centimeters; (b) at most 9 centimeters; (c) at least 2 and at most 7 centimeters? F(x)= -{8
4. An electronic system fails if both of its components fail. Let X be the time (in hours)until the system fails. Experience has shown thatWhat is the probability that the system lasts at least 200 but not more than 300 hours? t P(X > t) = (1 + 2) e 200 -t/200 t 0.
2. The mean and standard deviation in midterm tests of a probability course are 72 and 12, respectively. These quantities for final tests are 68 and 15. What final grade is comparable to Velma’s 82 in the midterm.
1. Mr. Norton owns two appliance stores. In store 1 the number of TV sets sold by a salesperson is, on average, 13 per week with a standard deviation of five. In store 2 the number of TV sets sold by a salesperson is, on average, seven with a standard deviation of four.Mr. Norton has a position
2. Let X be a random variable with E(X) = 3 and E(X − 3)(4 − X)= −15. Find Var(−3X + 8).
1. Two fair dice are tossed. Let X be the sum of the outcomes. Find Var(X) and σX.
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