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probability and stochastic modeling

Fundamentals Of Probability With Stochastic Processes 4th Edition Saeed Ghahramani - Solutions

16. 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)
15. 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.
14. 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.
13. For n = 1, 2, 3, . . . , let xn = (−1)n√n. Let X be a discrete random variable with theset of possible values A = {xn : n = 1, 2, 3, . . .} and probability mass functionShow that even thoughdoes not exist. == p(x) = P(X = x) = == 6 (TN)2
12. 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.
11. Let X be a random variable defined byLet Y be a random variable defined byWhich one of X and Y is more concentrated about 0 and why? P(X-1) = P(X = 1) = 1/2.
10. 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
9. Suppose that X is a discrete random variable with E(X) = 1 and EX(X − 2)= 3.Find Var(−3X + 5).
8. 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.)
7. Let X be the number of claims received, within a year, for an auto insurance policy offered by an insurance company. Let p, the probability mass function of X, be given byWhat percentage of the number of claims are within one standard deviation of the mean? n 25 30 35 40 45 50 55 p(n) 0.20 0.30
6. Let X be a discrete random variable with the set of possible values {x1, x2, . . . , xn};X is called a discrete uniform random variable if(a) Find E(X) and Var(X) for the special case, where P xi = i, 1 ≤ i ≤ n. Note that n i=1 i = n(n + 1)/2 and Pn i=1 i2 = n(n + 1)(2n + 1)/6.(b) Let X be a
5. Find the variance and the standard deviation of a random variable X with distribution function F(x)= 3/4 3/8
4. In the inventory of a multinational office supply retailing corporation, there are 7200 80-sheet smooth paper Renee pads made of acid free and ink-friendly paper in France.The retailing corporation sells these pads only in packs of 12. Orders received by the retailing corporation are from
3. Find the variance of X, the random variable with probability mass function p(x) = ((x - 3|+1)/28 x=-3, -2, -1, 0, 1, 2, 3 otherwise.
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,
Suppose that, for a discrete random variable X, E(X) = 2 and EX(X − 4)= 5. Find the variance and the standard deviation of −4X + 12.
What is the variance of the random variable X, the outcome of rolling a fair die?
4. Let F be the distribution function of a random variable X. Find E(X). 0 t
3. In a certain lottery, 15,000 tickets are sold. If there are 300 prizes of $5 each, 50 prizes of $50 each, and one grand prize of $1,000, what is the fair price for each ticket?
2. Two fair dice are tossed and the maximumof the outcomes is denoted byX. Find E(X).
1. Let X be a discrete random variable with probability mass functionFind E(2X). 0 1 1 3 7 p(x) 0.2 0.3 0.4 0.1
22. To an engineering class containing 2n − 3 male and three female students, there are n work stations available. To assign each work station 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
21. (a) Show thatis a probability mass function.(b) Let X be a random variable with probability mass function p given in part (a);find E(X). 1 p(n) = n 1, n(n+1)'
20. 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).
19. 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.
18. 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????Pc j=0 αj = 1. A child is selected at random from the set of all children in the world. Let this child
17. 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 P∞ i=1 iri =r/(1 −r)2,|r|
16. Let X be the number of different birthdays among four persons selected randomly. Find E(X).
15. IfX is a random number selected from the first 10 positive integers, what is the expected value of X(11 − X)?
14. The distribution function of a random variable X is given byCalculate E(X), E ????X2 − 2|X|, and E ????X|X|. 0 if x-3
13. The amount that an insurance policy pays for hospitalization is $a per day up to 4 days and $(a/2) thereafter. Let X be the number of days a randomly selected policy holder who needs hospitalization is hospitalized. If the probability mass function of X is given byfind the expected amount of
12. A box contains 10 disks of radii 1, 2, . . . , 10, respectively. What is the expected value of the circumference of a disk selected at random from this box?
11. (a) Show that p(x) =????|x| + 12/27, x = −2, −1, 0, 1, 2, is the probability mass function of a random variable X.(b) Calculate E(X), E????|X|, and E(2X2 − 5X + 7).
10. It is well known that P∞x=1 1/x2 = π2/6.(a) Show that p(x) = 6/(πx)2, x = 1, 2, 3, . . . is the probability mass function of a random variable X.(b) Prove that E(X) does not exist.
9. The demand for a certain weekly magazine at a newsstand is a random variable with probability mass function p(i) = (10 − i)/18, i = 4, 5, 6, 7. If the magazine sells for$a and costs $2a/3 to the owner, and the unsold magazines cannot be returned, how many magazines should be ordered every week
8. A box contains 20 fuses, of which five are defective. What is the expected number of defective items among three fuses selected randomly?
7. Let X be a discrete random variable with the following probability mass functionFind E(cosX). x /6 /4 /3 /2 p(x) 0.2 0.4 0.3 0.1
6. 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?
5. 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
4. 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?
3. Let X be a discrete random variable with probability mass functionFind E 2(X − 1)(3 − X). -2 0 2 4 p(x) 1/3 1/4 1/4 1/6
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
A box contains 10 disks of radii 1, 2, . . . , and 10, respectively. What is the expected value of the area of a disk selected at random from this box?
The probability mass function of a discrete random variable X is given byWhat is the expected value of X(6 − X)? p(x) = [x/15 x = 1,2,3,4,5 otherwise.
An urn contains w white and b blue chips. A chip is drawn at random and then is returned to the urn along with c > 0 chips of the same color.Prove that if n = 2, 3, 4, . . . , such experiments are made, then at each draw the probability of a white chip is still w/(w +b) and the probability of a
An urn contains w white and b blue chips. A chip is drawn at random and then is returned to the urn along with c > 0 chips of the same color. This experiment is then repeated successively. Let Xn be the number of white chips drawn during the first n draws.Show that E(Xn) = nw/(w + b).
The tanks of a country’s army are numbered 1 to N. In a war this country loses n random tanks to the enemy, who discovers that the captured tanks are numbered. If X1,X2, . . . ,Xn are the numbers of the captured tanks, what is E(maxXi)? How can the enemy use E(maxXi) to find an estimate of N, the
Let X0 be the amount of rain that will fall in the United States on the next Christmas day. For n > 0, let Xn be the amount of rain that will fall in the United States on Christmas n years later. Let N be the smallest number of years that elapse before we get a Christmas rainfall greater than X0.
In the lottery of a certain state, players pick six different integers between 1 and 49, the order of selection being irrelevant. The lottery commission then selects six of these numbers at random as the winning numbers. A player wins the grand prize of $1,200,000 if all six numbers that he has
A college mathematics department sends 8 to 12 professors to the annual meeting of the American Mathematical Society, which lasts five days. The hotel at which the conference is held offers a bargain rate of a dollars per day per person if reservations are made 45 or more days in advance, but
We write the numbers a1, a2, . . . , an on n identical balls and mix them in a box. What is the expected value of a ball selected at random?
We flip a fair coin twice and let X be the number of heads obtained.What is the expected value of X?
2. We choose 13 numbers at randomand without replacement fromthe set {1, 2, . . . , 100}.Let X be the median of the numbers selected. Find the probability mass function of X.Note that the median of the 13 numbers selected is the number in the middle when they are put in order. For example, if the
1. The number of claims filed with a car insurance company, per week, is a randomvariable with probability mass function(a) Find the probability of at least one such claim next week.(b) If we are given that there were no more than 5 claims filed with the company two weeks ago, find the probability
18. 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
17. 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
16. From a drawer that contains 10 pairs of gloves, six gloves are selected randomly. Let X be the number of pairs of gloves obtained. Find the probability mass function of X.
15. Let X be the number of vowels (not necessarily distinct) among the first five letters of a random arrangement of the following expression.ELIZABETHTAYLOR Find the probability mass function of X. Count the letter Y as a consonant
14. 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 −
13. 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
12. Suppose that the number of claims received by an insurance company in a given week is independent of the number of claims received in any other week. An actuary has calculated that the probability mass function of the number of claims received in a random week isFind the probability that the
11. Let X be the number of claims filed by a randomly selected customer under a certain homeowner’s insurance policy during a ten-year period. Suppose that an actuary has estimated that p, the probability mass function of X, satisfies p(n + 1) = 0.32p(n), n ≥ 0.What is the probability that a
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.
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, 3,
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 functionsrespectively. p(x) = (-), x= 1,2,3,..., and 4! 34-y q(y) y = 0, 1, 2, 3, 4, y! (4-y)!
4. The distribution function of a random variable X is given byDetermine the probability mass function of X and sketch its graph. 0 if x-2 1/2 if-2
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.
Let X be the number of births in a hospital until the first girl is born.Determine the probability mass function and the distribution function of X. Assume that the probability is 1/2 that a baby born is a girl.
In the experiment of rolling a balanced die twice, let X be the maximum of the two numbers obtained. Determine and sketch the probability mass function and the distribution function of X.
Can a function of the formbe a probability mass function? H p(x)= 0 () x = 1,2,3,... elsewhere
3. There are 50 students enrolled in a class, and they arrive one at a time, independently of each other. Suppose that the Xth student is the first one who shares his birthday with another student already present in the classroom. Find the probability mass function of X. Assume that the birth rates
2. Suppose that the distribution function of a random variable X is given byFind P(X > 3) and P(X > 5 | X > 3). F(t)= 1- t2 t
1. For what value of k, if any, is the function p(n) = k/n, n = 1, 2, 3, . . . , a probability mass function? (3 points)
20. Let the time until a new car breaks down be denoted by X, and letThen 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. Y = X 5 if X 5 if X > 5.
19. 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.
18. 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.
17. Let X be a randomly selected point from the interval (0, 3). What is the probability that X2 − 5X + 6 > 0?
16. 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?
15. 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 two-digit random numbers until we obtain one between 4
14. 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 randomtime.Suppose that X is the waiting time for this passenger; find the distribution function of X. Assume that seats are always
13. In the U.S., for fellowship, the Casualty Actuarial Society requires passing a series of nine rigorous exams taken in order plus certain other learning objectives. Suppose that the probability is p1 for a student to pass exam 1, and, for 2 ≤ i ≤ 9, if the student has passed exam i − 1,
12. Determine if the following is a distribution function. (1/2)et t
11. Determine if the following is a distribution function. t if t 0 F(t)= 1+t 0 ift < 0.
10. Determine if the following is a distribution function. e- if t0 F(t)= T ift < 0.
9. A random variable X is called symmetric about 0 if for all x ∈ R,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 1 − F(t).(c) P(X = t) = F(t) + F(−t) − 1.
8. Let X be a random variable with distribution function F. For p (0 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:Find Q0.50, called the median of F; Q0.25, called the first quartile of F; and Q0.75,
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:Suppose that this grocery store’s total sales of rice do not reach 600 kilograms on any given day.(a) Find the value of k.(b) What is the probability that the
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(a) Sketch the graph of F.(b) Calculate the following quantities: P(X 1/2), P(X = 3/2), and P(1 0 (1/4)t + 1/4 F(t) = 1/2 t
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
Suppose that a bus arrives at a station every day between 10:00 A.M. and 10:30 A.M., at random. Let X be the arrival time; find the distribution function of X and sketch its graph.
For the experiment of flipping a fair coin twice, let X be the number of tails and calculate F(t), the distribution function of X, and then sketch its graph.
A random number is selected from the interval (0, π/2). What is the probability that its sine is greater than its cosine?
The diameter of a flat metal disk manufactured by a factory is a random number between 4 and 4.5.What is the probability that the area of such a flat disk chosen at random is at least 4.41π?
10. Starting with Hillary, two players, Hillary and Donald, take turns and roll a fair die. The one who rolls a 6 first is the winner. What is the probability that Hillary will win?Hint: For i ≥ 1, let Ai be the event that the first 6 is rolled on Hillary’s ith turn. Let H be the event that
9. 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
8. Suppose that of the individuals who have been exposed to the dust product of the mineral asbestos, 3.3 per 1000 people develop mesothelioma. Suppose that Keith, a mine worker exposed to such a dust product, was tested for mesothelioma, and the test result came back positive. If 92% of the time

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