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probability and stochastic modeling

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

12. Let the joint probability density function of X and Y be given by f (x, y) = B 8xy if 0 ≤ x < y ≤ 1 0 otherwise. Determine if E(XY ) = E(X)E(Y ).
11. Let the joint probability density function of random variables X and Y be given by f (x, y) = B x2e−x(y+1) if x ≥ 0, y ≥ 0 0 elsewhere. Are X and Y independent? Why or why not?
independently have the same amount of cholesterol.
10. Suppose that the amount of cholesterol in a certain type of sandwich is 100X milligrams, where X is a random variable with the following density function: f (x) =    2x + 3 18 if 2 < x < 4 0 otherwise. Find the probability that two such sandwiches made
9. Let the joint probability density function of random variables X and Y be given by f (x, y) = B 2 if 0 ≤ y ≤ x ≤ 1 0 elsewhere. Are X and Y independent? Why or why not?
8. The joint probability mass function p(x, y) of the random variables X and Y is given by the following table. Determine if X and Y are independent. y x 0123 0 0.1681 0.1804 0.0574 0.0041 1 0.1804 0.1936 0.0616 0.0044 2 0.0574 0.0616 0.0196 0.0014 3 0.0041 0.0044 0.0014 0.0001
7. A fair coin is tossed n times by Adam and n times by Andrew. What is the probability that they get the same number of heads?
6. Let X and Y be two independent random variables with distribution functions F and G, respectively. Find the distribution functions of max(X, Y ) and min(X, Y ).
5. What is the probability that there are exactly two girls among the first seven and exactly four girls among the first 15 babies born in a hospital in a given week? Assume that the events that a child born is a girl or is a boy are equiprobable.
4. From an ordinary deck of 52 cards, eight cards are drawn at random and without replacement. Let X and Y be the number of clubs and spades, respectively. Are X and Y independent?
3. Let X and Y be independent random variables each having the probability mass function p(x) = 1 2 *2 3 ,x , x = 1, 2, 3, . . . . Find P (X = 1, Y = 3) and P (X + Y = 3).
2. Let the joint probability mass function of random variables X and Y be given by p(x, y) =    1 7 x2 y if (x, y) = (1, 1), (1, 2), (2, 1) 0 elsewhere. Are X and Y independent? Why or why not?
1. Let the joint probability mass function of random variables X and Y be given by p(x, y) =    1 25(x2 + y2 ) if x = 1, 2, y = 0, 1, 2 0 elsewhere. Are X and Y independent? Why or why not?
29. As Liu Wen from Hebei University of Technology in Tianjin, China, has noted in the April 2001 issue of The American Mathematical Monthly, in some reputable probability and statistics texts it has been asserted that “if a two-dimensional distribution function F (x, y) has a continuous density
28. For α > 0, β > 0, and γ > 0, the following function is called the bivariate Dirichlet probability density function f (x, y) = 0(α + β + γ ) 0(α)0(β)0(γ ) xα−1 yβ−1 (1 − x − y)γ −1 if x ≥ 0, y ≥ 0, and x + y ≤ 1; f (x, y) = 0, otherwise. Prove that fX, the marginal
27. Consider a disk centered at O with radius R. Suppose that n ≥ 3 points P1, P2, ... , Pn are independently placed at random inside the disk. Find the probability that all these points are contained in a closed semicircular disk. Hint: For each 1 ≤ i ≤ n, let Ai be the endpoint of the
26. Let X and Y be continuous random variables with joint probability density function f (x, y). Let Z = Y/X, X ,= 0. Prove that the probability density function of Z is given by fZ(z) = E ∞ −∞ |x|f (x, xz) dx.
25. A point is selected at random and uniformly from the region R = $ (x, y): |x|+|y| ≤ 1 % . Find the probability density function of the x-coordinate of the point selected at random.
24. Two points are placed on a segment of length $ independently and at random to divide the line into three parts. What is the probability that the length of none of the three parts exceeds a given value α, $/3 ≤ α ≤ $?
23. A farmer who has two pieces of lumber of lengths a and b (a
22. Two numbers x and y are selected at random from the interval (0, 1). For i = 0, 1, 2, determine the probability that the integer nearest to x + y is i. Note: This problem was given by Hilton and Pedersen in the paper “A Role for Untraditional Geometry in the Curriculum,” published in the
21. Three points M, N, and L are placed on a circle at random and independently. What is the probability that MNL is an acute angle?
20. Let g and h be two probability density functions with probability distribution functions G and H, respectively. Show that for −1 ≤ α ≤ 1, the function f (x, y) = g(x)h(y)! 1 + α[2G(x) − 1][2H (y) − 1] " is a joint probability density function of two random variables. Moreover, prove
19. Suppose that h is the probability density function of a continuous random variable. Let the joint probability density function of two random variables X and Y be given by f (x, y) = h(x)h(y), x ∈ R, y ∈ R. Prove that P (X ≥ Y ) = 1/2.
18. Let X and Y be random variables with finite expectations. Show that if P (X ≤ Y ) = 1, then E(X) ≤ E(Y ).
17. Two points X and Y are selected at random and independently from the interval (0, 1). Calculate P (Y ≤ X and X2 + Y 2 ≤ 1).
16. On a line segment AB of length $, two points C and D are placed at random and independently. What is the probability that C is closer to D than to A?
15. A farmer makes cuts at two points selected at random on a piece of lumber of length $. What is the expected value of the length of the middle piece?
14. A man invites his fiancée to an elegant hotel for a Sunday brunch. They decide to meet in the lobby of the hotel between 11:30 A.M. and 12 noon. If they arrive at random times during this period, what is the probability that the first to arrive has to wait at least 12 minutes?
13. Let R be the bounded region between y = x and y = x2. A random point (X, Y ) is selected from R. (a) Find the joint probability density function of X and Y . (b) Calculate the marginal probability density functions of X and Y . (c) Find E(X) and E(Y ).
12. Let X and Y have the joint probability density function f (x, y) = B 1 if 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 0 elsewhere. CalculateP (X+Y ≤ 1/2), P (X−Y ≤ 1/2), P (XY ≤ 1/4), andP (X2+Y 2 ≤ 1).
11. Let the joint probability density function of random variables X and Y be given by f (x, y) =    1 2 ye−x if x > 0, 0 < y < 2 0 elsewhere. Find the marginal probability density functions of X and Y .
10. Let the joint probability density function of random variables X and Y be given by f (x, y) = B 8xy if 0 ≤ y ≤ x ≤ 1 0 elsewhere. (a) Calculate the marginal probability density functions of X and Y , respectively. (b) Calculate E(X) and E(Y ).
9. Let the joint probability density function of random variables X and Y be given by f (x, y) = B 2 if 0 ≤ y ≤ x ≤ 1 0 elsewhere. (a) Calculate the marginal probability density functions of X and Y , respectively. (b) Find E(X) and E(Y ). (c) Calculate P (X < 1/2), P (X < 2Y ), and P (X = Y
8. From an ordinary deck of 52 cards, seven cards are drawn at random and without replacement. Let X and Y be the number of hearts and the number of spades drawn, respectively. (a) Find the joint probability mass function of X and Y . (b) Calculate P (X ≥ Y ).
7. In a community 30% of the adults are Republicans, 50% are Democrats, and the rest are independent. For a randomly selected person, let X = B 1 if he or she is a Republican 0 otherwise, Y = B 1 if he or she is a Democrat 0 otherwise. Calculate the joint probability mass function of X and Y .
6. Two dice are rolled. The sum of the outcomes is denoted by X and the absolute value of their difference by Y . Calculate the joint probability mass function of X and Y and the marginal probability mass functions of X and Y .
5. Thieves stole four animals at random from a farm that had seven sheep, eight goats, and five burros. Calculate the joint probability mass function of the number of sheep and goats stolen.
4. Let the joint probability mass function of discrete random variables X and Y be given by p(x, y) =    1 25(x2 + y2 ) if x = 1, 2, y = 0, 1, 2 0 otherwise. Find P (X > Y ), P (X + Y ≤ 2), and P (X + Y = 2).
3. Let the joint probability mass function of discrete random variables X and Y be given by p(x, y) = B k(x2 + y2) if (x, y) = (1, 1), (1, 3), (2, 3) 0 otherwise. Determine (a) the value of the constant k, (b) the marginal probability mass functions of X and Y , and (c) E(X) and E(Y ).
2. Let the joint probability mass function of discrete random variables X and Y be given by p(x, y) = B c(x + y) if x = 1, 2, 3, y = 1, 2 0 otherwise. Determine (a) the value of the constantc, (b) the marginal probability mass functions of X and Y , (c) P (X ≥ 2 | Y = 1), (d) E(X) and E(Y ).
1. Let the joint probability mass function of discrete random variables X and Y be given by p(x, y) =    k *x y , if x = 1, 2, y = 1, 2 0 otherwise. Determine (a) the value of the constant k, (b) the marginal probability mass functions of X and Y , (c) P (X > 1 | Y = 1), (d) E(X)
18. A beam of length $ is rigidly supported at both ends. Experience shows that whenever the beam is hit at a random point, it breaks at a position X units from the right end, where X/$ is a beta random variable. If E(X) = 3$/7 and Var(X) = 3$2/98, find P ($/7 < X < $/3).
17. Let X be a uniform random variable over the interval (1 − θ, 1 + θ), where 0 < θ < 1 is a given parameter. Find a function ofX, say g(X), so that E 4 g(X)5 = θ 2.
16. The number of phone calls to a specific exchange is a Poisson process with rate 23 per hour. Calculate the probability that the time until the 91st call is at least 4 hours.
15. The breaking strength of a certain type of yarn produced by a certain vendor is normal with mean 95 and standard deviation 11. What is the probability that, in a random sample of size 10 from the stock of this vendor, the breaking strengths of at least two are over 100?
14. Suppose that the weights of passengers taking an elevator in a certain building are normal with mean 175 pounds and standard deviation 22. What is the minimum weight for a passenger who outweighs at least 90% of the other passengers?
13. In a measurement, a number is rounded off to the nearest k decimal places. Let X be the rounding error. Determine the probability distribution function of X and its parameters.
12.. The number of minutes that a train from Milan to Rome is late is an exponential random variable X with parameter λ. Find P ! X > E(X)" .
11. The grades of students in a calculus-based probability course are normal with mean 72 and standard deviation 7. If 90, 80, 70, and 60 are the respective lowest, A, B, C, and D, what percent of students in this course get A’s, B’s, C’s, D’s, and F’s?
10. Determine the value(s) of k for which the following is a density function. f (x) = ke−x2+3x+2 , −∞ < x < ∞.
9. The time that it takes for a calculus student to answer all the questions on a certain exam is an exponential random variable with mean 1 hour and 15 minutes. If all 10 students of a calculus class are taking that exam, what is the probability that at least one of them completes it in less than
8. Let X be an exponential random variable with parameter λ. Prove that P (α ≤ X ≤ α + β) ≤ P (0 ≤ X ≤ β).
7. Suppose that the diameter of a randomly selected disk produced by a certain manufacturer is normal with mean 4 inches and standard deviation 1 inch. Find the distribution function of the diameter of a randomly chosen disk, in centimeters.
4. Let X, the lifetime of a light bulb, be an exponential random variable with parameter λ. Is it possible that X satisfies the following relation? P (X ≤ 2) = 2P (2 < X ≤ 3). If so, for what value of λ? 5. The time that it takes for a computer system to fail is exponential with mean 1700
3. One thousand random digits are generated. What is the probability that digit 5 is generated at most 93 times?
2. It is known that the weight of a random woman from a community is normal with mean 130 pounds and standard deviation 20. Of the women in that community who weigh above 140 pounds, what percent weigh over 170 pounds?
1. For a restaurant, the time it takes to deliver pizza (in minutes) is uniform over the interval (25, 37). Determine the proportion of deliveries that are made in less than half an hour.
2. One of the most popular distributions used to model the lifetimes of electric components is the Weibull distribution, whose probability density function is given by f (t) = αt α−1 e−tα , t > 0, α > 0. Determine for which values of α the hazard function of a Weibull random variable is
1. Experience shows that the failure rate of a certain electrical component is a linear function. Suppose that after two full days of operation, the failure rate is 10% per hour and after three full days of operation, it is 15% per hour. (a) Find the probability that the component operates for at
12. For an integer n ≥ 3, let X be a random variable with the probability density function f (x) = 0 *n + 1 2 , √nπ 0*n 2 , * 1 + x2 n ,−(n+1)/2 , −∞ < x < ∞. Such random variables have significant applications in statistics. They are called t-distributed with n degrees of freedom.
11. Prove that B(α, β) = 0(α)0(β) 0(α + β) .
10. For α, β > 0, show that B(α, β) = 2 E ∞ 0 t 2α−1 (1 + t 2 ) −(α+β) dt. Hint: Make the substitution x = t 2/(1 + t 2) in B(α, β) = E 1 0 xα−1 (1 − x)β−1 dx.
9. Under what conditions and about which point(s) is the probability density function of a beta random variable symmetric?
8. For complicated projects such as construction of spacecrafts, project managers estimate two quantities, a andb, the minimum and maximum lengths of time it will take for a project to be completed, respectively. In estimatingb, they consider all of the possible complications that might delay the
7. Suppose that while daydreaming, the fraction X of the time that one commits brave deeds is beta with parameters (5, 21). What is the probability that next time Jeff is daydreaming, he commits brave deeds at least 1/4 of the time?
6. At a certain university, the fraction of students who get a C in any section of a certain course is uniform over (0, 1). Find the probability that the median of these fractions for the 13 sections of the course that are offered next semester is at least 0.40.
5. The proportion of resistors a procurement office of an engineering firm orders every month, from a specific vendor, is a beta random variable with mean 1/3 and variance 1/18. What is the probability that next month, the procurement office orders at least 7/12th of its purchase from this vendor?
4. Suppose that new blood pressure medicines introduced are effective on 100p% of the patients, where p is a beta random variable with parameters α = 20 and β = 13. What is the probability that a new blood pressure medicine is effective on at least 60% of the hypertensive population?
3. For what value of c is the following a probability density function of some random variable X? Find E(X) and Var(X). f (x) = B cx4(1 − x)5 0 < x < 1 0 otherwise.
2. Is the following a probability density function? Why or why not? f (x) = B 120x2(1 − x)4 0 < x < 1 0 otherwise.
1. Is the following the probability density function of some beta random variableX? If so, find E(X) and Var(X). f (x) = B 12x(1 − x)2 0 < x < 1 0 otherwise.
10. 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 1 or 0. Suppose that the length of a character (in bits) is a geometric random variable with parameter p. Suppose that
9. Howard enters a bank that has n tellers. All the tellers are busy serving customers, and there is exactly one queue being served by all tellers, with one customer ahead of Howard waiting to be served. If the service time of a customer is exponential with parameter λ, find the distribution of
8. (a) Let Z be a standard normal random variable. Show that the random variable Y = Z2 is gamma and find its parameters. (b) Let X be a normal random variable with mean µ and standard deviation σ. Find the distribution function of W = *X − µ σ ,2 .
7. For n = 0, 1, 2, 3, . . . , calculate 0(n + 1/2).
6. A manufacturer produces light bulbs at a Poisson rate of 200 per hour. The probability that a light bulb is defective is 0.015. During production, the light bulbs are tested one by one, and the defective ones are put in a special can that holds up to a maximum of 25 light bulbs. On average, how
5. Customers arrive at a restaurant at a Poisson rate of 12 per hour. If the restaurant makes a profit only after 30 customers have arrived, what is the expected length of time until the restaurant starts to make profit?
4. Let f be the density function of a gamma random variable X, with parameters (r, λ). Prove that # ∞ −∞ f (x) dx = 1.
3. In a hospital, babies are born at a Poisson rate of 12 per day. What is the probability that it takes at least seven hours before the next three babies are born?
2. Let X be a gamma random variable with parameters (r, λ). Find the distribution function of cX, where c is a positive constant.
1. Show that the gamma density function with parameters (r, λ) has a unique maximum at (r − 1)/λ.
15. Prove that if X is a positive, continuous, memoryless random variable with distribution function F, then F (t) = 1 − e−λt for some λ > 0. This shows that the exponential is the only distribution on (0,∞) with the memoryless property\
14. Let X, the lifetime (in years) of a radio tube, be exponentially distributed with mean 1/λ. Prove that [X], the integer part of X, which is the complete number of years that the tube works, is a geometric random variable.
13. The random variable X is called double exponentially distributed if its density function is given by f (x) = ce−|x| , −∞ < x < +∞. (a) Find the value ofc. (b) Prove that E(X2n) = (2n)! and E(X2n+1) = 0.
12. 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 1 or 0. Suppose that L, the length of a character (in bits) is a geometric random variable with parameter p. If a
11. In a factory, a certain machine operates for a period which is exponentially distributed with parameter λ. Then it breaks down and will be in repair shop for a period, which is also exponentially distributed with mean 1/λ. The operating and the repair times are independent. For this machine,
10. Mr. Jones is waiting to make a phone call at a train station. There are two public telephone booths next to each other, occupied by two persons, say A and B. If the duration of each telephone call is an exponential random variable with λ = 1/8, what is the probability that among Mr. Jones, A,
9. The profit is $350 for each computer assembled by a certain person. Suppose that the assembler guarantees his computers for one year and the time between two failures of a computer is exponential with mean 18 months. If it costs the assembler $40 to repair a failed computer, what is the expected
8. Suppose that the time it takes for a novice secretary to type a document is exponential with mean 1 hour. If at the beginning of a certain eight-hour working day the secretary receives 12 documents to type, what is the probability that she will finish them all by the end of the day?
7. Suppose that, at an Italian restaurant, the time, in minutes, between two customers ordering pizza is exponential with parameter λ. What is the probability that (a) no customer orders pizza during the next t minutes; (b) the next pizza order is placed in at least t minutes but no later than s
5. Guests arrive at a hotel, in accordance with a Poisson process, at a rate of five per hour. Suppose that for the last 10 minutes no guest has arrived. What is the probability that (a) the next one will arrive in less than 2 minutes; (b) from the arrival of the tenth to the arrival of the
4. The time between the first and second heart attacks for a certain group of people is an exponential random variable. If 50% of those who have had a heart attack will have another one within the next five years, what is the probability that a person who had one heart attack five years ago will
3. Let X be an exponential random variable with mean 1. Find the probability density function of Y = − ln X.
2. Find the median of an exponential random variable with rate λ. Recall that for a continuous distribution F, the median Q0.5 is the point at which F (Q0.5) = 1/2.
1. Customers arrive at a postoffice at a Poisson rate of three per minute. What is the probability that the next customer does not arrive during the next 3 minutes?
36. Let I = # ∞ 0 e−x2/2 dx; then I 2 = E ∞ 0 8 E ∞ 0 e−(x2+y2)/2 dy9 dx. Let y/x = s and change the order of integration to show that I 2 = π/2. This gives an alternative proof of the fact that - is a distribution function. The advantage of this method is that it avoids polar
35. 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 positive real number. Find the expected value of the distance from a certain tree to its nearest neighbor.
34. At an archaeological site 130 skeletons are found and their heights are measured and found to be approximately normal with mean 172 centimeters and variance 81 centimeters. At a nearby site, five skeletons are discovered and it is found that the heights of exactly three of them are above 185
33. The amount of soft drink in a bottle is a normal random variable. Suppose that in 7% of the bottles containing this soft drink there are less than 15.5 ounces, and in 10% of them there are more than 16.3 ounces. What are the mean and standard deviation of the amount of soft drink in a randomly

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