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Fundamentals Of Probability 2nd Edition Saeed Ghahramani - Solutions
8 Let X be a continuous random variable with probability density function Find E(In X). [2/x if 1 < x
7 Let the probability density function of tomorrow's Celsius temperature be h. In terms of h, calculate the corresponding probability density function and its expectation for Fahrenheit temperature. Hint: Let C and F be tomorrow's temperature in Celsius and Fahrenheit, respectively. Then F = 1.8C
6 Let Y be a continuous random variable with probability distribution func- tion F(y) = - , where A, k, and a are positive constants. (Such distribution functions arise in the study of local computer network performance.) Find E(Y).)
5 Find the expected value of a random variable X with the density function [1/(1-x) if-1
4 A random variable X has the density function f(x)= (x) = {3e-) 3e- if 0x00 0 otherwise. Calculate E(ex).
3 The mean and standard deviation of the lifetime of a car muffler manu- factured by company A are 5 and 2 years, respectively. These quantities for car mufflers manufactured by company B are, respectively, 4 years and 18 months. Brian buys one muffler from company A and one from company B. That of
2 The time it takes for a student to finish an aptitude test (in hours) has the density function 6(x 1)(2x) if 1
1 The distribution function for the duration of a certain soap opera (in tens of hours) is 1- (16/x) if x 4 | F(x) = { 1- (a) Find E(X). (b) Show that Var(X) does not exist. if x < 4.
26 Show that if all three of n, N, and D so that n/N 0, D/N converges to a small number, and nD/N , then for all x, N D (P)(X-P) n-x N x! This formula shows that the Poisson distribution is the limit of the hyper- geometric distribution.
25 A farmer, plagued by insects, seeks to attract birds to his property by distributing seeds over a wide area. Let be the average number of seeds per unit area, and suppose that the seeds are distributed in a way that the probability of having any number of seeds in a given area depends only on
24 Suppose that n babies were born at a county hospital last week. Also suppose that the probability of a baby having blonde hair is p. If k of these n babies are blondes, what is the probability that the ith baby born is blonde?
23 Experience shows that 75% of certain kinds of seeds germinate when planted under normal conditions. Determine the minimum number of seeds to be planted so that the chance of at least five of them germinating - is more than 90%.
22 A bowl contains w white and b blue chips. Chips are drawn at random and with replacement until a blue chip is drawn. What is the probability that (a) exactly n draws are required; (b) at least n draws are required?
21 The probability that a child of a certain family inherits a certain disease is 0.23 independently of other children inheriting the disease. If the family has five children and the disease is detected in one child, what is the probability that exactly two more children have the disease as well?
20 From the set {x:0 < x < 1) numbers are selected at random and inde- pendently and rounded to three decimal places. What is the probability that 0.345 is obtained (a) for the first time after 1000 selections; (b) for the third time after 3000 selections?
19 An insurance company claims that only 70% of the drivers regularly use seat belts. In a statistical survey, it was found that out of 20 randomly selected drivers, 12 regularly used seat belts. Is this sufficient evidence to conclude that the insurance company's claim is false?
18 Suppose that 6% of the claims received by an insurance company are for damages from vandalism. What is the probability that at least three of the 20 claims received on a certain day are for vandalism damages?
17 A certain type of seed when planted fails to germinate with probability 0.06. If 40 of such seeds are planted, what is the probability that at least 36 of them germinate?
16 A bowl contains 10 red and six blue chips. What is the expected number of blue chips among five randomly selected chips?
15 Of the 28 professors in a certain department, 18 drive foreign and 10 drive domestic cars. If five of these professors are selected at random, what is the probability that at least three of them drive foreign cars?
14 Past experience shows that 30% of the customers entering Harry's Clothing Store will make a purchase. Of the customers who make a purchase, 85% use credit cards. Let X be the number of the next six customers who enter the store, make a purchase, and use a credit card. Find the probability
13 In data communication, one method for error control makes the receiver send a positive acknowledgment for every message received with no de- tected error, and a negative acknowledgment for all the others. Suppose that (i) acknowledgments are transmitted error free, (ii) a negative ac-
12 What is the probability that the sixth toss of a die is the first 6?
11 A fair coin is tossed successively until a head occurs. If N is the number of tosses required, what are the expected value and the variance of N?
10 The policy of the quality control division of a certain corporation is to reject a shipment if more than 5% of its items are defective. A shipment of 500 items is received, 30 of them are randomly tested, and two have been found defective. Should that shipment be rejected?
9 Suppose that a certain bank returns bad checks at a Poisson rate of three per day. What is the probability that this bank returns at most four bad checks during the next two days?
8 Suppose that 10 trains arrive independently at a station every day, each at a random time between 10:00 AM. and 11:00 AM.. What is the ex- pected number and the variance of those that arrive between 10:15 A.M. and 10:28 A.M.?
7 From a panel of prospective jurors, 12 are selected at random. If there are 200 men and 160 women on the panel, what is the probability that more than half of the jury selected are women?
6 In a community, the chance of a set of triplets is 1 in 1000 births. Determine the probability that the second set of triplets in this community occurs before the 2000th birth.
5 A doctor has five patients with migraine headaches. He prescribes for all five a drug that relieves the headaches of 82% of such patients. What is the probability that the medicine will not relieve the headaches of two of these patients?
4 A university has n students, 70% of whom will finish an aptitude test in less than 25 minutes. If 12 students are selected at random, what is the probability that at most two of them will not finish in 25 minutes?
3 A restaurant serves eight fish entres, 12 beef, and 10 poultry. If customers select from these entres randomly, what is the expected number of fish entres ordered by the next four customers?
2 The time between the arrival of two consecutive customers at a postoffice is 3 minutes, on average. Assuming that customers arrive in accordance with a Poisson process, find the probability that tomorrow during the lunch hour (between noon and 12:30 PM.) fewer than seven customers arrive.
1 Of police academy applicants, only 25% will pass all the examinations. Suppose that 12 successful candidates are needed. What is the probability that, by examining 20 candidates, the academy finds all of the 12 persons needed?
27 Suppose that, from a box containing D defective and N-D nondefective items, n (D) are drawn one-by-one, at random and without replacement. (a) Find the probability that the kth item drawn is defective. (b) If the (k-1)st item drawn is defective, what is the probability that the kth item is also
26 To estimate the number of trout in a lake, we caught 50 trout, tagged and returned them. Later we caught 50 trout and found that four of them were tagged. From this experiment estimate n, the total number of trout in the lake. Hint: Let p be the probability of four tagged trout among the 50
25 Adam rolls a well-balanced die until he gets a 6. Andrew rolls the same die until he rolls an odd number. What is the probability that Andrew rolls the die more than Adam does?
24 In the Banach matchbox problem, Example 5.23, find the probability that the box which is emptied first is not the one that is first found empty.
23 In the Banach matchbox problem, Example 5.23, find the probability that when the first matchbox is emptied (not found empty) there are exactly m matches in the other box.
22 Twelve hundred eggs, of which 200 are rotten, are distributed randomly in 100 cartons, each containing a dozen eggs. These cartons are then sold to a restaurant. How many cartons should we expect the chef of the restaurant to open before finding one without rotten eggs?
21 In data communication, messages are usually combinations of characters, and each character consists of a number of bits. A bit is the smallest unit of information and is either I or 0. Suppose that the length of a character (in bits) is a geometric random variable with parameter p. Furthermore,
20 Let X be a geometric random variable with parameter p. and n and m be nonnegative integers. (a) Prove that P(X>n+m | X>m) = P(X >n). What is the meaning of this property in terms of Bernoulli trials? This is called the memoryless property of geometric random variables. (b) For what values of n
19 On average, how many independent games of poker are required until a preassigned player is dealt a straight? (See Exercise 20 of Section 2.4 for a definition of a straight. The cards have distinct consecutive values that are not of the same suit: for example, 3 of hearts, 4 of hearts, 5 of
18 A fair coin is flipped repeatedly. What is the probability that the fifth tail occurs before the tenth head?
17 A computer network consists of several stations connected by various me- dia (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 two
16 A vending machine contains cans of grapefruit juice that cost 75 cents each, but it is not working properly. The probability that it accepts a coin is 10%. Angela has a quarter and five dimes. Determine the probability that she should try the coins at least 50 times before she gets a can of
15 people are selected at random and without replacement, what is the probability that at least two persons have contributed? 15 The probability is p that a message sent over a communication channel is garbled. If the message is sent repeatedly until it is transmitted reliably, and if each time it
14 In an annual charity drive, 35% of a population of 560 make contribu- tions. If, in a statistical survey,
13 Florence is moving and wishes to sell her package of 100 computer diskettes. Unknown to her, 10 of those diskettes are defective. Sharon will purchase them if a random sample of 10 contains no more than one defective disk. What is the probability that she buys them?
population.
12 Suppose that 15% of the population of a town are senior citizens. Let X be the number of nonsenior citizens who enter a mall before the tenth senior citizen arrives. Find the probability function of X. Assume that each customer who enters the mall is a random person from the entire
11 Solve the Banach matchbox problem (Example 5.23) for the case where the matchbox in the right pocket contains M matches, the matchbox in his left pocket contains N matches, and m min(M, N).
10 On average, how many games of bridge are necessary before a player is dealt three aces? A bridge hand is 13 randomly selected cards from an ordinary deck of 52 cards.
9 The digits after the decimal point of a random number between 0 and 1 are numbers selected at random, with replacement, independently, and successively from the set (0, 1, ..., 9). In a random number from (0, 1), on the average, how many digits are there before the fifth 3?
8 In rural Ireland, a century ago, the students had to form a line. The student at the front of the line would be asked to spell a word. If he spelled it correctly, he was allowed to sit down. If not, he received a whack on the hand with a switch and was sent to the end of the line. Suppose that a
7 Suppose that independent Bernoulli trials with parameter p are performed successively. Let N be the number of trials needed to get x successes, and X be the number of successes in the first n trials. Show that x P(N =n) = P(X = x). n Note: By this relation, in coin tossing, for example, we can
6 The probability is p that a randomly chosen lightbulb is defective. We screw a bulb into a lamp and switch on the current. If the bulb works, we stop; otherwise, we try another and continue until a good bulb is found. What is the probability that at least n bulbs are required?
5 A store has 50 lightbulbs available for sale. Of these, five are defective. A customer buys eight light bulbs randomly from this store. What is the probability that he finds exactly one defective light bulb among them?
4 A certain basketball player makes a foul shot with probability 0.45. What is the probability that (a) his first basket occurs on the sixth shot; (b) his first and second baskets occur on his fourth and eighth shots, respectively?
3 Suppose that 20% of a group of people have hazel eyes. What is the probability that the eighth passenger boarding a plane is the third one having hazel eyes? Assume that passengers boarding the plane form a randomly chosen group.
2 The probability is p that Marty hits target M when he fires at it. The probability is q that Alvie hits target A when he fires at it. Marty and Alvie fire one shot each at their targets. If both of them hit their targets, they stop; otherwise, they will continue. (a) What is the probability that
1 An absentminded professor does not remember which of his 12 keys will open his office door. If he tries them at random and with replacement: (a) On average, how many keys should he try before his door opens? (b) What is the probability that he opens his office door after only three tries?
probable
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 function and the probability distribution functions of X. Assume that in a three-child family all gender distributions are equally
9 Experience shows that X, the number of customers entering a postoffice, during any period of length, is a random variable whose probability function is of the form p(i) = k(2n (a) Determine the value of k. i = 0, 1, 2,.... (b) Compute P(X 1).
8 The fasting blood-glucose levels of 30 children are as follows. 58 62 80 58 64 76 80 80 80 58 62 64 76 76 58 64 62 80 58 58 80 64 58 58 62 62 76 76 62 64 80 62.76 Let X be the fasting blood-glucose level of a child chosen randomly from this group. Find the distribution function of X.
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 probability distributions of X and Y are as follows:
6 The annual amount of rainfall (in centimeters) in a certain area is a random variable with distribution function F(x)= x
5 A professor has made 30 exams of which eight are difficult, 12 are reason- able, and 10 are easy. The exams are mixed up, and the professor selects four of them at random to give to four sections of the course he is teaching. How many sections would be expected to get a difficult test?
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 that P(X > 1) = (1 + 2010) e-1/200 10. What is the probability that the system lasts at least 200, but not more than 300 hours?
3 A statistical survey shows that only 2% of secretaries know how to use the highly sophisticated word processor language TEX. If a certain mathe- matics department prefers to hire a secretary who knows TEX, what is the least number of applicants that should be interviewed so as to have at least a
2 A word is selected at random from the following poem of the Persian poet and mathematician Omar Khayyam (1048-1131), translated by Edward Fitzgerald (1808-1883), an English poet. Find the expected value of the length of the word.
1 An urn contains 10 chips numbered from 0 to 9. Two chips are drawn at random and without replacement. What is the probability function of their total?
2, each for one week. The salesperson in store 1 sold 10 sets, and the salesperson in store 2 sold six sets. Based on this information, which person should Mr. Norton hire? 2 The mean and standard deviation in midterm tests of a probability course are 72 and 12, respectively. These quantities for
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
15 Let X and Y be two discrete random variables with the identical set of possible values A = (a, az, an), wherea, a... a,, are n different real numbers. Show that if E(X') = E(Y'), r=1, 2, 1, then X and Y are identically distributed. That is, P(X = 1) = P(Y=1) forta1, a2. a.
14 Let X and Y be two discrete random variables with the identical set of possible values A = {a,b, c), wherea, b, and c are three different real numbers. Show that if E(X) = E(Y) and Var(X) = Var(Y), then X and Y are identically distributed. That is, P(X = 1) = P(Y=1) fort =a, b, c.
13 Let X be a discrete random variable; let 0 < s
12 Let X be a discrete random variable, and let n 1 be an integer. Show that if the nth moment of X exists, then the (n - 1)st moment of X also exists. Hence, if the nth moment of X exists, then all preceding moments of X exist as well.
11 For n=1, 2, 3, ..., let x = (-1)"n. Let X be a discrete random variable with the set of possible values A = {x,n = 1, 2, 3,...) and probability function p(x) = P(X = x) = 6n2 Show that even though xp(x)
10 A drunken man has n keys, one of which opens the door to his office. He tries the keys at random, one by one, and independently. Compute the mean and the variance of the number of trials required to open the door if the wrong keys (a) are not eliminated; (b) are eliminated.
9 Let X be a random variable defined by P(X = 1) = P(X = 1) =Let Y be a random variable defined by P(Y-10) P(Y = 10) Which one of X and Y is more concentrated about 0 and why? B
8 In a game, Emily gives Harry three well-balanced quarters to flip. Harry will get to keep all the ones that will land heads. He will return those landing tails. However, if all three coins land tails, Harry must pay Emily two dollars. Find the expected value and the variance of Harry's net gain.
7 Suppose that X is a discrete random variable with E(X) = 1 and EX(X-2)] 3. Find Var(-3X+5).
6 What are the expected number, the variance, and the standard deviation of the number of spades in a poker hand? (A poker hand is a set of five cards that are randomly selected from an ordinary deck of 52 cards.)
5 Let X be a random integer from the set (1. 2. 3..... N). Find E(X), Var(X), and x.
4 Find the variance and the standard deviation of a random variable X with distribution function 0 x < 3 3/8 -3x0 F(x)= 3/4 0x6 1 x 6.
3 Find the variance of X, the random variable with probability function P(x) = {(x-3+1)/28x=-3, -2, -1, 0, 1, 2, 3 10 otherwise.
2 The temperature of a material is measured by two devices. Using the first device, the expected temperature is t with standard deviation 0.8; using the second device, the expected temperature is 7 with standard deviation. 0.3. Which device measures the temperature more precisely?
1 Mr. Jones is about to purchase a business. There are two businesses avail- able. The first has a daily expected profit of $150 with standard deviation $30, and the second has a daily expected profit of $150 with standard de- viation $55. If Mr. Jones is interested in a business with a steady
19 Adam and three of his friends are playing bridge. (a) If, in holding certain hand, Adam announces that he has a king, what is the probability that he has at least one more king? (b) If, for some other hand, Adam announces that he has the king of diamonds, what is the probability that he has at
18 A student at a certain university will pass the oral Ph.D. qualifying ex- amination if at least two of the three examiners pass her or him. Past experience shows that (a) 15% of the students who take the qualifying exam are not prepared, and (b) each examiner will independently pass 85% of the
17 Solve the following problem, from the "Ask Marilyn" column of Parade Magazine, August 9, 1992. Three of us couples are going to Lava Hot Springs next week- end. We're staying two nights, and we've rented two studios, because each holds a maximum of only four people. One couple will get their own
16 A child is lost at Epcot Center in Florida. The father of the child believes that the probability of his being lost in the east wing of the center is 0.75, and in the west wing 0.25. The security department sends three officers to the east wing and two to the west to look for the child. Suppose
15. An urn contains nine red and one blue balls. A second urn contains one red and five blue balls. One ball is removed from each urn at random and without replacement, and all of the remaining balls are put into a third urn. What is the probability that a ball drawn randomly from the third urn is
14 Urn I contains 25 white and 20 black balls. Urn II contains 15 white and 10 black balls. An urn is selected at random and one of its balls is drawn randomly and observed to be black and then returned to the same urn. If a second ball is drawn at random from this urn, what is the probability that
13 Six fair dice are tossed independently. Find the probability that the number of 1's minus the number of 2's will be 3.
12 An experiment consists of first tossing an unbiased coin and then rolling a fair die. If we perform this experiment successively, what is the probability of obtaining a heads on the coin before a 1 or 2 on the die?
11 Urns I and II contain three pennies and four dimes, and two pennies and five dimes, respectively. One coin is selected at random from each urn. If exactly one of them is a dime, what is the probability that the coin selected from urn I is the dime?
10 Suppose that 10 dice are thrown and we are told that among them at least one has landed 6. What is the probability that there are two or more sixes?
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