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probability and stochastic modeling

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

22. Prove the following generalization of Exercise 21: n m m Cov (a; Xi, b;Y;) = a;b, Cov(X;, Y;). - i=1 j=1 i=1 j=1
21. Show that for random variables X, Y, Z, andW and constantsa, b,c, and d, Cov(aX + bY, cZ + dW)= ac Cov(X,Z) + bc Cov(Y,Z) + ad Cov(X,W) + bd Cov(Y,W).Hint: For a simpler proof, use the results of Exercise 6.
20. Let S be the sample space of an experiment. Let A and B be two events of S. Let IA and IB be the indicator variables for A and B. That is,Show that IA and IB are positively correlated if and only if P(A | B) > P(A), and if and only if P(B | A) > P(B). 1 IA(w)= if WE A if w A,
19. Let X be a random variable. Prove that Var(X) = mint E(X − t)2.Hint: Let μ = E(X) and look at the expansion of E(X − t)2= E(X − μ + μ − t)2.
18. Let X and Y be jointly distributed with joint probability density functionDetermine if X and Y are positively correlated, negatively correlated, or uncorrelated.Hint: Note that for all a > 0, R ∞0 xne−ax dx = n!/an+1. f(x,y) = 12 -ry-r if x > 0, y >0 otherwise.
17. Find the variance of a sum of n randomly and independently selected points from the interval (0, 1).
16. Let X and Y have the following joint probability density function(a) Calculate Var(X + Y ).(b) Show that X and Y are not independent. Explain why this does not contradict Exercise 27 of Section 8.2. 8xy if 0
15. A voltmeter is used to measure the voltage of voltage sources, such as batteries. Every time this device is used, a random error is made, independent of other measurements, with mean 0 and standard deviation σ. Suppose that we want to measure the voltages, V1 and V2, of two batteries. If a
14. Let X and Y be independent random variables with expected values μ1 and μ2, and variances σ2 1 and σ2 2, respectively. Show that Var(XY)=0+
13. Mr. Jones has two jobs. Next year, he will get a salary raise of X thousand dollars from one employer and a salary raise of Y thousand dollars from his second. Suppose that X and Y are independent random variables with probability density functions f and g, respectively, whereWhat are the
8. For random variables X and Y, show that Cov(X + Y,X − Y ) = Var(X) − Var(Y ).9. Prove that Var(X − Y ) = Var(X) + Var(Y ) − 2 Cov(X, Y ).10. Let X and Y be two independent random variables.(a) Show that X − Y and X + Y are uncorrelated if and only if Var(X) =Var(Y ).(b) Show that
5. In n independent Bernoulli trials, each with probability of success p, let X be the number of successes and Y the number of failures. Calculate E(XY ) and Cov(X, Y ).
4. Thieves stole four animals at random from a farm that had seven sheep, eight goats, and five burros. Calculate the covariance of the number of sheep and goats stolen.
3. Roll a balanced die and let the outcome be X. Then toss a fair coin X times and let Y denote the number of tails. Find Cov(X, Y ) and interpret the result.Hint: Let p(x, y) be the joint probability mass function of X and Y . To save time, use the table for p(x, y) constructed in Example 8.2.
2. Let the joint probability mass function of random variables X and Y be given byFind Cov(X, Y ). p(x,y) = 0 x(x+y) if x = 1,2,3, y=3,4 elsewhere.
1. Ann cuts an ordinary deck of 52 cards and displays the exposed card. After Ann places her stack back on the deck, Andy cuts the same deck and displays the exposed card.Counting jack, queen, and king as 11, 12, and 13, let X and Y be the numbers on the cards that Ann and Andy expose,
Using relation (10.10), calculate the variance of a negative binomial random variable X, with parameter (r, p).
Let X be the lifetime of an electronic system and Y be the lifetime of one of its components. Suppose that the electronic system fails if the component does (but not necessarily vice versa). Furthermore, suppose that the joint probability density function of X and Y (in years) is given by(a)
Ann cuts an ordinary deck of 52 cards and displays the exposed card. Andy cuts the remaining stack of cards and displays his exposed card. Counting jack, queen, and king as 11, 12, and 13, let X and Y be the numbers on the cards that Ann and Andy expose, respectively. Find Cov(X, Y ) and interpret
There are 300 cards in a box numbered 1 through 300. Therefore, the number on each card has one, two, or three digits. A card is drawn at random from the box. Suppose that the number on the card has X digits of which Y are 0. Determine whether X and Y are positively correlated, negatively
2. Let X be the number of students in Dr. Brown-Rose’s English 101 who will fail the course next semester. Show thatHint: Letand apply the Cauchy–Schwarz inequality P(X = 0) < Var(X) E(X2)
1. Shiante does not remember in which one of her 11 disk storage wallets she stored the last DVD that she purchased. Determine the expected number of the wallets that she should search to find the DVD. Assume that, in her search, she chooses the wallets randomly.Hint: For 1 ≤ i ≤ 11, let Xi = 1
19. Under what condition does Cauchy–Schwarz’s inequality become equality?
18. From an urn that contains a large number of red and blue chips, mixed in equal proportions, 10 chips are removed one by one and at random. The chips that are removed before the first red chip are returned to the urn. The first red chip, together with all those that follow, is placed in another
17. Let {X1,X2, . . .} be a sequence of continuous, independent, and identically distributed random variables. LetFind E(N). N = min{n: X> X2 X3 >> Xn-1, Xn-1
16. Let X and Y be nonnegative random variables with an arbitrary joint distribution function.Let(a) Show that(b) By calculating expected values of both sides of part (a), prove thatNote that this is a generalization of the result explained in Remark 6.4. I(x, y) == if X, Y> y otherwise.
15. From an ordinary deck of 52 cards, cards are drawn at random, one by one, and without replacement until a heart is drawn. What is the expected value of the number of cards drawn?Hint: See Exercise 13, Section 3.2.
14. There are 25 students in a probability class. What is the expected number of the days of the year that are birthdays of at least two students? Assume that the birthrates are constant throughout the year and that each year has 365 days.
13. There are 25 students in a probability class. What is the expected number of birthdays that belong only to one student? Assume that the birthrates are constant throughout the year and that each year has 365 days.Hint: Let Xi = 1 if the birthday of the ith student is not the birthday of any
12. Suppose that 80 balls are placed into 40 boxes at random and independently.What is the expected number of the empty boxes?
11. A coin is tossed n times (n > 4). What is the expected number of exactly three consecutive heads?Hint: Let E1 be the event that the first three outcomes are heads and the fourth outcome is tails. For 2 ≤ i ≤ n−3, let Ei be the event that the outcome (i−1) is tails, the outcomes i, (i
10. Let {X1,X2, . . . ,Xn} be a set of independent random variables with P(Xj = i) = pi (1 ≤ j ≤ n and i ≥ 1). Let hk =P∞ i=k pi. Using Theorem 10.2, prove that E[min(X1, X2,..., Xn)] = h k=1
9. Solve the following problem posed by Michael Khoury, U.S. Mathematics Olympiad Member, in “The Problem Solving Competition,” Oklahoma Publishing Company and the American Society for Communication of Mathematics, February 1999.Bob is teaching a class with n students. There are n desks in the
8. Let X1, X2, . . . , Xn be positive, identically distributed random variables. For 1 ≤ i ≤ n, letShow that Z1, Z2, . . . , Zn are also identically distributed and, for 1 ≤ i ≤ n, find E(Zi). Xi Z X + X2 + + Xn
7. A cultural society is arranging a party for its members. The cost of a band to play music, the amount that the caterer will charge, the rent of a hall to give the party, and other expenses (in dollars) are uniform random variables over the intervals (1300, 1800),(1800, 2000), (800, 1200), and
6. An absentminded professor wrote n letters and sealed them in envelopes without writing the addresses on the envelopes. Having forgotten which letter he had put in which envelope, he wrote the n addresses on the envelopes at random. What is the expected number of the letters addressed
5. A company puts five different types of prizes into their cereal boxes, one in each box and in equal proportions. If a customer decides to collect all five prizes, what is the expected number of the boxes of cereals that he or she should buy?
4. Let the joint probability density function of random variables X and Y beFind E(X), E(Y ), and E(X2 + Y 2). f(x, y): 2e-(x+2y) if x 0, y 0 === 0 otherwise.
3. Let X, Y, and Z be three independent random variables such that E(X) = E(Y ) =E(Z) = 0, and Var(X) =Var(Y ) =Var(Z) = 1. Calculate E[X2(Y + 5Z)2].
2. A calculator is able to generate random numbers from the interval (0, 1). We need five random numbers from (0, 2/5). Using this calculator, how many independent random numbers should we generate, on average, to find the five numbers needed?
1. Let the probability density function of a random variable X be given byFind E(X2 + X). f(x)= | 1| if 0 x 2 otherwise.
A box contains nine light bulbs, of which two are defective. What is the expected value of the number of light bulbs that one will have to test (at random and without replacement) to find both defective bulbs?
Dr. Windler’s secretary accidentally threw a patient’s file into the wastebasket.A few minutes later, the janitor cleaned the entire clinic, dumped the wastebasket containing the patient’s file randomly into one of the seven garbage cans outside the clinic, and left.Determine the expected
Exactly n married couples are living in a small town. What is the expected number of intact couples after m deaths occur among the couples? Assume that the deaths occur at random, there are no divorces, and there are no new marriages.
A well-shuffled ordinary deck of 52 cards is divided randomly into four piles of 13 each. Counting jack, queen, and king as 11, 12, and 13, respectively, we say that a match occurs in a pile if the jth card is j. What is the expected value of the total number of matches in all four piles?
A die is rolled 15 times. What is the expected value of the sum of the outcomes?
5. Suppose that 20% of the physicians working for a certain hospital retire before age 65, 30% retire at ages 65-69, and the remaining 50% retire at age 70 or later. If physicians retire independently of each other, what is the probability that the median age of five randomly selected retired
4. A system has 7 components, and it functions if and only if at least one of its components functions. Suppose that the lifetimes of the components are independent, identically distributed exponential random variables with mean 1 year. Furthermore, suppose that currently all the components are
3. For what value of c is the following a joint probability density function of four random variables X, Y, Z, and T? For that value of c find P(X x>0, y> 0, 2 > 0, t>0 (1+x+y+z+1) 6 f(x, y, z, t) = otherwise.
2. Let X1 be a random point from the interval (0, 1), X2 be a random point from the interval (0,X1), X3 be a random point from the interval (0,X2), · · · , and Xn be a random point from the interval (0,Xn−1). Find the joint probability density function of X1, X2, . . . , Xn.Hint: Use relation
1. Let (X, Y,Z) be a point selected at random in the unit sphere(x, y, z) : x2 + y2 + z2 ≤ 1;that is, the sphere of radius 1 centered at the origin. [Note that the volume of a sphere with radius R is (4/3)πR3.](a) Findf, the joint probability density function of X, Y, and Z.(b) Find the joint
10. Let X1, X2, and X3 be independent random variables from (0, 1). Find the probability density function and the expected value of the midrange of these random variables[X(1) + X(3)]/2.
8. A system consists of n components whose lifetimes form an independent sequence of random variables. Suppose that the system functions as long as at least one of its components functions. Let F1, F2, . . . , Fn be the distribution functions of the lifetimes of the components of the system. In
7. A system consists of n components whose lifetimes form an independent sequence of random variables. In order for the system to function, all components must function. Let F1, F2, . . . , Fn be the distribution functions of the lifetimes of the components of the system. In terms of F1, F2, . . .
6. Alvie, a marksman, fires seven independent shots at a target. Suppose that the probabilities that he hits the bull’s-eye, he hits the target but not the bull’s-eye, and he misses the target are 0.4, 0.35, and 0.25, respectively. What is the probability that he hits the bull’s-eye three
4. The joint probability density function of random variables X, Y, and Z is given by(a) Determine the value of c.(b) Find P(X (c(x + y +22) if 0x, y, z 1 f(x, y, z) = 10 otherwise.
3. Suppose that n points are selected at random and independently inside the cube=(x, y, z) : − a ≤ x ≤a, −a ≤ y ≤a, −a ≤ z ≤ a.Find the probability that the distance of the nearest point to the center is at least r (r < a).
2. Let X be the smallest number obtained in rolling a balanced die n times. Calculate the distribution function and the probability mass function of X.
1. An urn contains 100 chips of which 20 are blue, 30 are red, and 50 are green. Suppose that 20 chips are drawn at random and without replacement. Let B, R, and G be the number of blue, red, and green chips, respectively. Calculate the joint probability mass function of B, R, and G.
2. Of the emails that arrive in Samantha’s inbox, 45% are personal, 40% are work-related, and 15% are unsolicited commercial emails. Samantha logs into her email account and finds that she has 25 new emails. What is the probability that exactly 10 of them are work-related and at most 4 of them
1. Suppose that 40% of the students joining the mathematics department of a certain university major in actuarial science, 35% major in statistics and operation research, and 25% major in pure mathematics. There are 30 new students entering this department next fall. What is the probability that
8. Customers enter a department store at the rate of three per minute, in accordance with a Poisson process. If 30% of them buy nothing, 20% pay cash, 40% use charge cards, and 10% write personal checks, what is the probability that in five operating minutes of the store, five customers use charge
7. Suppose that the ages of 30% of the teachers of a country are over 50, 20% are between 40 and 50, and 50% are below 40. In a random committee of 10 teachers from this country, two are above 50.What is the probability mass function of those who are below 40?
6. Suppose that 50% of the watermelons grown on a farm are classified as large, 30% as medium, and 20% as small. Joanna buys five watermelons at random from this farm.What is the probability that (a) at least two of them are large; (b) two of them are large, two are medium, and one is small; (c)
5. Of the drivers who are insured by a certain insurance company and get into at least one accident during a random year, 15% are low-risk drivers, 35% are moderate-risk drivers, and 50% are high-risk drivers. Suppose that drivers insured by this company get into an accident independently of each
4. At a certain college, 16% of the calculus students get A’s, 34% B’s, 34% C’s, 14% D’s, and 2% F’s. What is the probability that, of 15 calculus students selected at random, five get B’s, five C’s, two D’s, and at least two A’s?
3. Suppose that each day the price of a stock moves up 1/8 of a point with probability 1/4, remains the same with probability 1/3, and moves down 1/8 of a point with probability 5/12. If the price fluctuations from one day to another are independent, what is the probability that after six days the
2. An urn contains 100 chips of which 20 are blue, 30 are red, and 50 are green. We draw 20 chips at random and with replacement. Let B, R, and G be the number of blue, red, and green chips, respectively. Calculate the joint probability mass function of B, R, and G.
1. Light bulbs manufactured by a certain factory last a random time between 400 and 1200 hours.What is the probability that, of eight such bulbs, three burn out before 550 hours, two burn out after 800 hours, and three burn out after 550 but before 800 hours?
Let X1, X2, . . . , Xr (r ≥ 4) have the joint multinomial probability mass function p(x1, x2, . . . , xr) with parameters n and p1, p2,. . . , pr. Find the marginal probability mass functions pX1 and pX1,X2,X3 .
A warehouse contains 500 TV sets, of which 25 are defective, 300 are in working condition but used, and the rest are brand new.What is the probability that, in a random sample of five TV sets from this warehouse, there are exactly one defective and exactly two brand new sets?
In a certain town, at 8:00 P.M., 30% of the TV viewing audience watch the news, 25% watch a certain comedy, and the rest watch other programs.What is the probability that, in a statistical survey of seven randomly selected viewers, exactly three watch the news and at least two watch the comedy?
2. All that we know about a horse race that was held last week in Louisville, Kentucky, is that all of the five horses that were competing reached the finish line, independently, at random times after 2:10 P.M. and before 2:22 P.M.(a) Find the probability that the winning horse passed the finish
1. Find the expected value of the distance between two random points selected independently from the interval (0, 1).
11. Let X1,X2, . . . ,Xn be n independently randomly selected points from the interval(0, θ), θ > 0. Prove thatwhere R = X(n) − X(1) is the range of these points.Hint: Use part (a) of Exercise 10. Also compare this with Exercise 21, Section 9.1. E(R) n-1 -0. n+1
10. Let X1,X2, . . . ,Xn be a random sample of size n from a population with continuous distribution function F and probability density function f.(a) Calculate the probability density function of the sample range, R =X(n) − X(1).(b) Use (a) to find the probability density function of the sample
9. Let X1 and X2 be two independent N(0, σ2) random variables. Find E[X(1)].Hint: Let f12(x, y) be the joint probability density function of X(1) and X(2). The desired quantity is RR xf12(x, y) dx dy, where the integration is taken over an appropriate region.
8. Let X1 and X2 be two independent exponential random variables each with parameterλ. Show that X(1) and X(2) − X(1) are independent.
7. Prove that G, the distribution function of [X(1) + X(n)]2, the midrange of a random sample of size n from a population with continuous distribution function F and probability density functionf, is given byHint: Use Theorem 9.6 to find f1n; then integrate over the region x + y ≤ 2t and x ≤ y.
6. Let X1, X2, X3, . . . , Xm be a sequence of nonnegative, independent binomial random variables, each with parameters (n, p). Find the probability mass function of X(i), 1 ≤ i ≤ m.
5. Let X1, X2, X3, . . . , Xn be a sequence of nonnegative, identically distributed, and independent random variables. Let F be the distribution function of Xi, 1 ≤ i ≤ n.Prove thatHint: Use Remark 6.4. E[|X (n)] = (1 [F(x)]) dx. 0
4. Let X1, X2, X3, and X4 be independent exponential random variables, each with parameter λ. Find P(X(4) ≥ 3λ).
3. A box contains 20 identical balls numbered 1 to 20. Seven balls are drawn randomly and without replacement. Find the probability mass function of the median of the numbers on the balls drawn.
The distance between two towns, A and B, is 30 miles. If three gas stations are constructed independently at randomly selected locations between A and B, what is the probability that the distance between any two gas stations is at least 10 miles?
Suppose that a machine consists of n components with the lifetimes X1, X2,. . . , Xn, respectively, where Xi’s are independent and identically distributed. Suppose that the machine remains operative unless k or more of its components fail. Then X(k), the kth order statistic of {X1,X2, . . . ,Xn},
Suppose that customers arrive at a warehouse from n different locations. Let Xi, 1 ≤ i ≤ n, be the time until the arrival of the next customer from location i; then X(1) is the arrival time of the next customer to the warehouse.
2. Let X, Y, and Z be continuous random variables with the joint probability density function given byFind P(X 2y e-(2x+y+2) x>0, y>0, 2>0 f(x, y, z) = otherwise.
1. In the inventory of a pharmacy, in a carton, there are 40 boxes of painkillers of which 15 are brand A, 10 are brand B, 6 are brand C, 4 are brand D, and 5 are brand E. An assistant pharmacist chooses 20 of these boxes randomly to move them inside the store for the over-the-counter sale. Let X,
26. (Roots of Cubic Equations) Solve the following exercise posed by S. A. Patil and D. S. Hawkins, Tennessee Technological University, Cookeville, Tennessee, in The College Mathematics Journal, September 1992.Let A, B, and C be independent random variables uniformly distributed on [0, 1]. What is
25. (Roots of Quadratic Equations) Three numbers A, B, and C are selected at random and independently from the interval (0, 1). Determine the probability that the quadratic equation Ax2 + Bx + C = 0 has real roots. In other words, what fraction of“all possible quadratic equations” with
24. A point is selected at random from the pyramid V =(x, y, z) : x, y, z ≥ 0, x + y + z ≤ 1.Letting (X, Y,Z) be its coordinates, determine if X, Y, and Z are independent.Hint: Recall that the volume of a pyramid is Bh/3, where h is the height and B is the area of the base.
23. Suppose that h is the probability density function of a continuous random variable. Let the joint probability density function of X, Y, and Z be f(x, y, z) = h(x)h(y)h(z), x, y, z ∈ R.Prove that P(X < Y < Z) = 1/6.
22. Let X1,X2, . . . ,Xn be n independent random numbers from (0, 1), and Yn = n · min(X1,X2, . . . ,Xn).Prove that lim n→∞P(Yn > x) = e−x, x ≥ 0.
21. Let X1,X2, . . . ,Xn be n independent random numbers from the interval (0, 1). Find E????max 1≤i≤n Xiand E????min 1≤i≤n Xi.
20. Let F be a distribution function. Prove that the functions Fn and 1 − (1 − F)n are also distribution functions.Hint: Let X1,X2, . . . ,Xn be independent random variables each with the distribution function F. Find the distribution functions of the random variables max(X1,X2, . . . ,Xn) and
19. (Reliability of Systems) To transfer water from point A to point B, a water-supply system with five water pumps located at the points 1, 2, 3, 4, and 5 is designed as in Figure 9.5. Suppose that whenever the system is turned on for water to flow from A to B, pump i, i ≤ 5, functions with
18. (Reliability of Systems) Consider the system whose structure is shown in Figure 9.4. Find the reliability of this system. 1 2 3 4 5 6 7 Figure 9.4 A diagram for the system of Exercise 18.
17. Suppose that the lifetimes of a certain brand of transistor are identically distributed and independent random variables with distribution function F. These transistors are randomly selected, one at a time, and their lifetimes are measured. Let the Nth be the first transistor that will last
16. An item has n parts, each with an exponentially distributed lifetime with mean 1/λ. If the failure of one part makes the item fail, what is the average lifetime of the item?Hint: Use the result of Example 9.4.
15. (Reliability of Systems) Suppose that a system functions if and only if at least k (1 ≤ k ≤ n) of its components function. Furthermore, suppose that pi = p for 1 ≤ i ≤ n. Find the reliability of this system. (Such a system is said to be a k-out-of-n system.)

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