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probability and stochastic modeling

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

13. 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?
12. On average, how many times should Ernie play poker in order to be dealt a straight flush (royal flush included)? (See Exercise 20 of Section 2.4 for definitions of a royal and a straight flush.)
11. 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.
10. 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?
9. 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?
8. 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?
7. 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.
6. 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?
5. 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?
4. 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}.
3. A graduate class consists of six students. What is the probability that exactly three of them are born either in April or in October?
2. The probability that a randomly selected person is female is 1/2. What is the expected number of the girls in the first-grade classes of an elementary school that has 64 first graders? What is the expected number of females in a family with six children?
1. From an ordinary deck of 52 cards, cards are drawn at random and with replacement. What is the probability that, of the first eight cards drawn, four are spades?
11. (The Clock Solitaire) An ordinary deck of 52 cards is well shuffled and dealt face down into 13 equal piles. The first 12 piles are arranged in a circle like the numbers on the face of a clock. The 13th pile is placed at the center of the circle. Play begins by turning over the bottom card in
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 probability distribution functions of X. Assume that in a three-child family all gender distributions are
9. Experience shows that X, the number of customers entering a post office, during any period of length t, is a random variable the probability mass function of which is of the form p(i) = k (2t)i i! , i = 0, 1, 2, . . . . (a) Determine the value of k. (b) Compute P (X < 4) and 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 62 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: x 15.85 15.9 16 16.1 16.2 P (X = x) 0.15
6. The annual amount of rainfall (in centimeters) in a certain area is a random variable with the distribution function F (x) = B 0 x < 5 1 − (5/x2) x ≥ 5. What 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
5. A professor has prepared 30 exams of which 8 are difficult, 12 are reasonable, 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 > t) = * 1 + t 200 , e−t/200, t ≥ 0. 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 mathematics 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 Persian poet and mathematician Omar Khayy¯am (1048–1131), translated by English poet Edward Fitzgerald (1808–1883). Find the expected value of the length of the word. The moving finger writes and, having writ, Moves on; nor all your
1. An urn contains 10 chips numbered from 0 to 9. Two chips are drawn at random and without replacement. What is the probability mass function of their total?
14. Let X and Y be two discrete random variables with the identical set of possible values A = {a1, a2, . . . , an}, where a1, a2, . . . , an are n different real numbers. Show that if E(Xr ) = E(Y r ), r = 1, 2,...,n − 1, then X and Y are identically distributed. That is, P (X = t) = P (Y = t)
13. 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 = t) = P (Y = t) for t =a, b, c.
12. Let X be a discrete random variable; let 0 < s < r. Show that if the rth absolute moment of X exists, then the absolute moment of order s of X also exists.
11. For n = 1, 2, 3,... , let xn = (−1)n√n. Let X be a discrete random variable with the set of possible values A = {xn : n = 1, 2, 3,...} and probability mass function p(xn) = P (X = xn) = 6 (πn)2 . Show that even though /∞ n=1 x3 n p(xn) < ∞, E(X3) does not exist.
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) = 1/2. Let Y be a random variable defined by P (Y = −10) = P (Y = 10) = 1/2. Which one of X and Y is more concentrated about 0 and why?
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
7. Suppose thatX is a discrete random variable withE(X) = 1 andE 4 X(X−2) 5 = 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, . . . , N}. Find E(X), Var(X), and σX.
4. Find the variance and the standard deviation of a random variable X with distribution function F (x) =    0 x < −3 3/8 −3 ≤ x < 0 3/4 0 ≤ x < 6 1 x ≥ 6.
3. Find the variance of X, the random variable with probability mass function p(x) = B! |x − 3| + 1 " /28 x = −3, −2, −1, 0, 1, 2, 3 0 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 t 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 available. 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 deviation $55. If Mr. Jones is interested in a business with a steady income,
19. To an engineering class containing 2n − 3 male and three female students, there are n work stations available. To assign each workstation to two students, the professor forms n teams one at a time, each consisting of two randomly selected students. In this process, let X be the number of
18. (a) Show that p(n) = 1 n(n + 1) , n ≥ 1, is a probability mass function. (b) Let X be a random variable with probability mass function p given in part (a); find E(X).
17. Suppose that n random integers are selected from {1, 2, . . . , N} with replacement. What is the expected value of the largest number selected? Show that for large N the answer is approximately nN/(n + 1).
16. An ordinary deck of 52 cards is well-shuffled, and then the cards are turned face up one by one until an ace appears. Find the expected number of cards that are face up.
15. Suppose that there exist N families on the earth and that the maximum number of children a family has isc. For j = 0, 1, 2, . . . ,c, let αj be the fraction of families with j children !/c j=0 αj = 1 " . A child is selected at random from the set of all children in the world. Let this child
14. A newly married couple decides to continue having children until they have one of each sex. If the events of having a boy and a girl are independent and equiprobable, how many children should this couple expect? Hint: Note that /∞ i=1 iri = r/(1 − r)2, |r| < 1.
13. Let X be the number of different birthdays among four persons selected randomly. Find E(X).
12. If X is a random number selected from the first 10 positive integers, what is E 4 X(11 − X)5 ?
11. The distribution function of a random variable X is given by F (x) =    0 if x < −3 3/8 if −3 ≤ x < 0 1/2 if 0 ≤ x < 3 3/4 if 3 ≤ x < 4 1 if x ≥ 4. Calculate E(X), E ! X2 − 2|X| " , and E ! X|X| " .
10. A box contains 10 disks of radii 1, 2, . . . , 10, respectively. What is the expected value of the circumference of a disk selected at random from this box?
9. (a) Show that p(x) = ! |x| + 1 "2 /27, x = −2, −1, 0, 1, 2, is the probability mass function of a random variable X. (b) Calculate E(X), E ! |X| " , and E(2X2 − 5X + 7).
8. It is well known that /∞ x=1 1/x2 = π2/6. (a) Show that p(x) = 6/(πx)2, x = 1, 2, 3,... is the probability mass function of a random variable X. (b) Prove that E(X) does not exist.
7. The demand for a certain weekly magazine at a newsstand is a random variable with probability mass function p(i) = (10−i)/18, i = 4, 5, 6, 7. If the magazine sells for $a and costs $2a/3 to the owner, and the unsold magazines cannot be returned, how many magazines should be ordered every week
6. A box contains 20 fuses, of which five are defective. What is the expected number of defective items among three fuses selected randomly?
5. An urn contains five balls, two of which are marked $1, two $5, and one $15. A game is played by paying $10 for winning the sum of the amounts marked on two balls selected randomly from the urn. Is this a fair game?
4. In a lottery, a player pays $1 and selects four distinct numbers from 0 to 9. Then, from an urn containing 10 identical balls numbered from 0 to 9, four balls are drawn at random and without replacement. If the numbers of three or all four of these balls matches the player’s numbers, he wins
3. In a lottery every week, 2,000,000 tickets are sold for $1 apiece. If 4000 of these tickets pay off $30 each, 500 pay off $800 each, one ticket pays off $1,200,000, and no ticket pays off more than one prize, what is the expected value of the winning amount for a player with a single ticket?
2. In a certain part of downtown Baltimore parking lots charge $7 per day. A car that is illegally parked on the street will be fined $25 if caught, and the chance of being caught is 60%. If money is the only concern of a commuter who must park in this location every day, should he park at a lot or
1. There is a story about Charles Dickens (1812–1870), the English novelist and one of the most popular writers in the history of literature. It is known that Dickens was interested in practical applications of mathematics. On the final day in March during a year in the second half of the
16. To an engineering class containing 23 male and three female students, there are 13 work stations available. To assign each work station to two students, the professor forms 13 teams one at a time, each consisting of two randomly selected students. In this process, let X be the total number of
15. A fair die is tossed successively. Let X denote the number of tosses until each of the six possible outcomes occurs at least once. Find the probability mass function of X. Hint: For 1 ≤ i ≤ 6, let Ei be the event that the outcome i does not occur during the first n tosses of the die. First
14. From a drawer that contains 10 pair of gloves, six gloves are selected randomly. Let X be the number of pairs of gloves obtained. Find the probability mass function of X. Count the letter Y as a consonant.
13. Let X be the number of vowels (not necessarily distinct) among the first five letters of a random arrangement of the following expression. Find the probability mass function of X.
12. Every Sunday, Bob calls Liz to see if she will play tennis with him on that day. If Liz has not played tennis with Bob since i Sundays ago, the probability that she will say yes to him is i/k, k ≥ 2, i = 1, 2, ... , k. Therefore, if, for example, Liz does not play tennis with Bob for k − 1
11. A binary digit or bit is a zero or one. A computer assembly language can generate independent random bits. Let X be the number of independent random bits to be generated until both 0 and 1 are obtained. Find the probability mass function of X·
10. In successive rolls of a fair die, let X be the number of rolls until the first 6 appears. Determine the probability mass function and the distribution function of X.
9. Let p(x) = 3/4(1/4)x , x = 0, 1, 2, 3,... , be probability mass function of a random variable X. Find F, the distribution function of X, and sketch its graph.
8. From 18 potential women jurors and 28 potential men jurors, a jury of 12 is chosen at random. Let X be the number of women selected. Find the probability mass function of X.
7. For each of the following, determine the value(s) of k for which p is a probability mass function. Note that in parts (d) and (e), n is a positive integer. (a) p(x) = kx, x = 1, 2, 3, 4, 5. (b) p(x) = k(1 + x)2, x = −2, 0, 1, 2. (c) p(x) = k(1/9)x , x = 1, 2, 3,... . (d) p(x) = kx, x = 1, 2,
6. A value i is said to be the mode of a discrete random variable X if it maximizes p(x), the probability mass function of X. Find the modes of random variables X and Y with probability mass functions p(x) = *1 2 ,x , x = 1, 2, 3, . . . , and q(y) = 4! y!(4 − y)! *1 4 ,y*3 4 ,4−y , y = 0, 1, 2,
5. Let X be the number of random numbers selected from {0, 1, 2, . . . , 9} independently until 0 is chosen. Find the probability mass functions of X and Y = 2X+1.
4. The distribution function of a random variable X is given by F (x) =    0 if x < −2 1/2 if −2 ≤ x < 2 3/5 if 2 ≤ x < 4 8/9 if 4 ≤ x < 6 1 if x ≥ 6. Determine the probability mass function of X and sketch its graph.
3. In the experiment of rolling a balanced die twice, let X be the sum of the two numbers obtained. Determine the probability mass function of X.
2. In the experiment of rolling a balanced die twice, let X be the minimum of the two numbers obtained. Determine the probability mass function and the distribution function of X and sketch their graphs.
1. Let p(x) = x/15, x = 1, 2, 3, 4, 5 be probability mass function of a random variable X. Determine F, the distribution function of X, and sketch its graph.
19. Let the time until a new car breaks down be denoted by X, and let Y = B X if X ≤ 5 5 if X > 5. Then Y is the life of the car, if it lasts less than 5 years, and is 5 if it lasts longer than 5 years. Calculate the distribution function of Y in terms of F, the distribution function of X.
18. In the United States, the number of twin births is approximately 1 in 90. At a certain hospital let X be the number of births until the first twins are born. Find the first quartile, the median, and the third quartile of X. See Exercise 8 for the definitions of these quantities.
17. Let X be a random point selected from the interval (0, 1). Calculate F, the distribution function of Y = X/(1 + X), and sketch its graph.
16. Let X be a randomly selected point from the interval(0, 3). What is the probability that X2 − 5X + 6 > 0?
15. In a small town there are 40 taxis, numbered 1 to 40. Three taxis arrive at random at a station to pick up passengers. What is the probability that the number of at least one of the taxis is less than 5?
14. A scientific calculator can generate two-digit random numbers. That is, it can choose a number at random from the set {00, 01, 02, . . . , 99}. To obtain a random number from the set {4, 5, . . . , 18}, show that we have to keep generating twodigits random numbers until we obtain one between 4
13. Airline A has commuter flights every 45 minutes from San Francisco airport to Fresno. A passenger who wants to take one of these flights arrives at the airport at a random time. Suppose that X is the waiting time for this passenger; find the distribution function of X. Assume that seats are
12. Determine if the following is a distribution function. F (t) = B (1/2)et t < 0 1 − (3/4)e−t t ≥ 0.
11. Determine if the following is a distribution function. F (t) =    t 1 + t if t ≥ 0 0 if t < 0.
10. Determine if the following is a distribution function. F (t) =    1 − 1 π e−t if t ≥ 0 0 if t < 0.
9. A random variable X is called symmetric about 0 if for all x ∈ R , P (X ≥ x) = P (X ≤ −x). Prove that if X is symmetric about 0, then for all t > 0 its distribution function F satisfies the following relations: (a) P ! |X| ≤ t " = 2F (t) − 1. (b) P ! |X| > t" = 2 4 1 − F (t)5 . (c)
8. Let X be a random variable with distribution function F. For p (0 < p < 1), Qp is said to be a quantile of order p if F (Qp−) ≤ p ≤ F (Qp). In a certain country, the rate at which the price of oil per gallon changes from one year to another has the following distribution function: F (x) =
7. A grocery store sells X hundred kilograms of rice every day, where the distribution of the random variable X is of the following form: F (x) =    0 x < 0 kx2 0 ≤ x < 3 k(−x2 + 12x − 3) 3 ≤ x < 6 1 x ≥ 6. Suppose that this grocery store’s total
6. From families with three children a family is chosen at random. Let X be the number of girls in the family. Calculate and sketch the distribution function of X. Assume that in a three-child family all gender distributions are equally probable.
5. F, the distribution function of a random variable X, is given by F (t) =    0 t < −1 (1/4)t + 1/4 −1 ≤ t < 0 1/2 0 ≤ t < 1 (1/12)t + 7/12 1 ≤ t < 2 1 t ≥ 2. (a) Sketch the graph of F. (b) Calculate the following quantities: P
4. The side measurement of a plastic die, manufactured by factory A, is a random number between 1 and 11 4 centimeters. What is the probability that the volume of a randomly selected die manufactured by this company is greater than 1.424? Assume that the die will always be a cube.
3. In a society of population N, the probability is p that a person has a certain rare disease independently of others. Let X be the number of people who should be tested until a person with the disease is found, X = 0 if no one with the disease is found. What are the possible values of X?
2. From an urn that contains five red, five white, and five blue chips, we draw two chips at random. For each blue chip we win $1, for each white chip we win $2, but for each red chip we lose $3. If X represents the amount that we either win or we lose, what are the possible values of X and
1. Two fair dice are rolled and the absolute value of the difference of the outcomes is denoted by X. What are the possible values of X, and the probabilities associated with them?
20. Adam and three of his friends are playing bridge. (a) If, holding a 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
19. A student at a certain university will pass the oral Ph.D. qualifying examination 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
18. Solve the following problem, asked of Marilyn Vos Savant in the “Ask Marilyn” column of Parade Magazine, August 9, 1992.Three of us couples are going to Lava Hot Springs next weekend. We’re staying two nights, and we’ve rented two studios, because each holds a maximum of only four
17. 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
16. A fair coin is tossed. If the outcome is heads, a red hat is placed on Lorna’s head. If it is tails, a blue hat is placed on her head. Lorna cannot see the hat. She is asked to guess the color of her hat. Is there a strategy that maximizes Lorna’s chances of guessing correctly? Hint:
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
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.

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