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

8. 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 ## xf12(x, y) dx dy, where the integration is taken over an appropriate region.
7. Let X1 and X2 be two independent exponential random variables each with parameter λ. Show that X(1) and X(2) − X(1) are independent.
6. Prove that G, the probability distribution function of [X(1)+X(n)] = 2, the midrange of a random sample of size n from a population with continuous probability distribution function F and probability density function f , is given by G(t) = n E t −∞ 4 F (2t − x) − F (x)5n−1 f (x) dx.
5. 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.
4. Let X1, X2, X3, ... , Xn be a sequence of nonnegative, identically distributed, and independent random variables. Let F be the probability distribution function of Xi, 1 ≤ i ≤ n. Prove that E[X(n)] = E ∞ 0 ! 1 − 4 F (x)5n" dx. Hint: Use Theorem 6.2.
3. Let X1, X2, X3, and X4 be independent exponential random variables, each with parameter λ. Find P (X(4) ≥ 3λ).
2. Two random points are selected from (0, 1) independently. Find the probability that one of them is at least three times the other.
1. Let X1, X2, X3, and X4 be four independently selected random numbers from (0, 1). Find P (1/4 < X(3) < 1/2).
25. (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
24. (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
23. 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.
22. (Reliability of Systems) To transfer water from point A to point B, a watersupply 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
21. 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.
20. 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.
19. Let F be a probability distribution function. Prove that the functions Fn and 1 − (1 − F )n are also probability distribution functions. Hint: Let X1, X2, . . . , Xn be independent random variables each with the probability distribution function F. Find the probability distribution
18. Let X1, X2, . . . , Xn be n independent random numbers from the interval (0, 1). Find E ! max 1≤i≤n Xi " and E ! min 1≤i≤n Xi " .
17. (Reliability of Systems) Consider the system whose structure is shown in Figure 9.4. Find the reliability of this system.
16. Suppose that the lifetimes of a certain brand of transistor are identically distributed and independent random variables with probability 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
15. 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 Exercise 13.
14. (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.)
13. Let X1, X2, . . . , Xn be independent exponential random variables with means 1/λ1, 1/λ2, ... , 1/λn, respectively. Find the probability distribution function of X = min(X1, X2, . . . , Xn).
replacement dies. He stops using the radio when the second replacement of the transistor goes out of order. Assuming that Arnold repairs the radio if it fails for any other reason, find the probability that he uses the radio at least 15 years.
12. Suppose that the lifetimes of radio transistors are independent exponential random variables with mean five years. Arnold buys a radio and decides to replace its transistor upon failure two times: once when the original transistor dies and once when the
11. Is the following a joint probability density function? f (x1, x2, . . . , xn) = B e−xn if 0 < x1 < x2 < ··· < xn 0 otherwise.
10. A point is selected at random from the cube # = $ (x, y, z): − a ≤ x ≤a, −a ≤ y ≤a, −a ≤ z ≤ a % . What is the probability that it is inside the sphere inscribed in the cube?
9. Inside a circle of radius R, n points are selected at random and independently. Find the probability that the distance of the nearest point to the center is at least r.
8. (a) Show that the following is a joint probability density function. f (x, y, z) =    −ln x xy if 0 < z ≤ y ≤ x ≤ 1 0 otherwise. (b) Suppose that f is the joint probability density function of X, Y , and Z. Find fX,Y (x, y) and fY (y).
7. Let the joint probability distribution function of X, Y , and Z be given by F (x, y, z) = (1 − e−λ1x )(1 − e−λ2y )(1 − e−λ3z ), x, y, z > 0, where λ1, λ2, λ3 > 0. (a) Are X, Y , and Z independent? (b) Find the joint probability density function of X, Y , and Z. (c) Find P (X <
6. Let X, Y , and Z be jointly continuous with the following joint probability density function: f (x, y, z) = B x2e−x(1+y+z) if x, y, z > 0 0 otherwise. Are X, Y , and Z independent? Are they pairwise independent?
5. From the set of families with two children a family is selected at random. Let X1 = 1 if the first child of the family is a girl; X2 = 1 if the second child of the family is a girl; and X3 = 1 if the family has exactly one boy. For i = 1, 2, 3, let Xi = 0 in other cases. Determine if X1, X2, and
4. Let the joint probability density function of X, Y , and Z be given by f (x, y, z) = B 6e−x−y−z if 0 < x < y < z < ∞ 0 elsewhere. (a) Find the marginal joint probability density function of X, Y ; X, Z; and Y , Z. (b) Find E(X).
3. Let p(x, y, z) = (xyz)/162, x = 4, 5, y = 1, 2, 3, and z = 1, 2, be the joint probability mass function of the random variables X, Y , Z. (a) Calculate the joint marginal probability mass functions of X, Y ; Y , Z; and X, Z. (b) Find E(Y Z).
2. A jury of 12 people is randomly selected from a group of eight Afro-American, seven Hispanic, three Native American, and 20 white potential jurors. Let A, H, N, and W be the number of Afro-American, Hispanic, Native American, and white jurors selected, respectively. Calculate the joint
1. From an ordinary deck of 52 cards, 13 cards are selected at random. Calculate the joint probability mass function of the numbers of hearts, diamonds, clubs, and spades selected.
22. (The Wallet Paradox) Consider the following “paradox” given by Martin Gardner in his book Aha! Gotcha (W. H. Freeman and Company, New York, 1981). Each of two persons places his wallet on the table. Whoever has the smallest amount of money in his wallet, wins all the money in the other
21. Let the joint probability density function of random variables X and Y be given by f (x, y) = B 1 if |y| < x, 0 < x < 1 0 otherwise. Show that E(Y | X = x) is a linear function of x while E(X | Y = y) is not a linear function of y.
20. There are prizes in 10% of the boxes of a certain type of cereal. Let X be the number of boxes of such cereal that Kim should buy to find a prize. Let Y be the number of additional boxes of such cereal that she should purchase to find another prize. Calculate the joint probability mass function
19. A bar of length $ is broken into three pieces at two random spots. What is the probability that the length of at least one piece is less than $/20?
18. If F is the probability distribution function of a random variable X, is G(x, y) = F (x) + F (y) a joint probability distribution function?
17. Let X and Y be two independent uniformly distributed random variables over the intervals (0, 1) and (0, 2), respectively. Find the probability density function of X/Y .
16. A point is selected at random from the bounded region between the curves y = x2 − 1 and y = 1 − x2. Let X be the x-coordinate, and let Y be the y-coordinate of the point selected. Determine if X and Y are independent.
15. Let the joint probability density function of X and Y be given by f (x, y) = B cx(1 − x) if 0 ≤ x ≤ y ≤ 1 0 otherwise. (a) Determine the value ofc. (b) Determine if X and Y are independent.
14. From an ordinary deck of 52 cards, cards are drawn successively and with replacement. Let X and Y denote the number of spades in the first 10 cards and in the second 15 cards, respectively. Calculate the joint probability mass function of X and Y .
13. Let X and Y be continuous random variables with the joint probability density function f (x, y) = B e−y if y > 0, 0 < x < 1 0 elsewhere. Find E(Xn | Y = y), n ≥ 1.
12. For # = $ (x, y): 0 < x + y < 1, 0 < x < 1, 0 < y < 1 % , a region in the plane, let f (x, y) = B 3(x + y) if (x, y) ∈ # 0 otherwise be the joint probability density function of the random variables X and Y . Find the marginal probability density functions of X and Y , and P (X + Y > 1/2).
11. Let the joint probability distribution function of the lifetimes of two brands of lightbulb be given by F (x, y) = B (1 − e−x2 )(1 − e−y2 ) if x > 0, y > 0 0 otherwise. Find the probability that one lightbulb lasts more than twice as long as the other.
10. A fair coin is flipped 20 times. If the total number of heads is 12, what is the expected number of heads in the first 10 flips?
9. Three concentric circles of radii r1, r2, and r3, r1 > r2 > r3, are the boundaries of the regions that form a circular target. If a person fires a shot at random at the target, what is the probability that it lands in the middle region?
8. Prove that the following cannot be the joint probability distribution function of two random variables X and Y . F (x, y) = B 1 if x + y ≥ 1 0 if x + y < 1.
7. Let X and Y have the joint probability density function below. Determine if E(XY ) = E(X)E(Y ). f (x, y) =    3 4 x2 y + 1 4 y if 0 < x < 1 and 0 < y < 2 0 elsewhere.
6. Let the joint probability density function of X and Y be given by f (x, y) =    c x if 0 < y < x, 0 < x < 2 0 elsewhere. (a) Determine the value ofc. (b) Find the marginal probability density functions of X and Y .
5. Calculate the probability mass function of the number of spades in a random bridge hand that includes exactly four hearts and three clubs.
4. Suppose that three cards are drawn at random from an ordinary deck of 52 cards. If X and Y are the numbers of diamonds and clubs, respectively, calculate the joint probability mass function of X and Y .
3. Calculate the probability mass function of the number of spades in a random bridge hand that includes exactly four hearts.
2. A fair die is tossed twice. The sum of the outcomes is denoted by X and the largest value by Y . (a) Calculate the joint probability mass function of X and Y ; (b) find the marginal probability mass functions of X and Y ; (c) find E(X) and E(Y ).
1. The joint probability mass function of X and Y is given by the following table. x y 123 2 0.05 0.25 0.15 4 0.14 0.10 0.17 6 0.10 0.02 0.02 (a) Find P (XY ≤ 6). (b) Find E(X) and E(Y ).
10. Let X and Y be independent (strictly positive) exponential random variables each with parameter λ. Are the random variables X + Y and X/Y independent?
9. Let X and Y be independent (strictly positive) gamma random variables with parameters (r1, λ) and (r2, λ), respectively. Define U = X + Y and V = X/(X + Y ). (a) Find the joint probability density function of U and V . (b) Prove that U and V are independent. (c) Show that U is gamma and V is
8. Prove that if X and Y are independent standard normal random variables, then X + Y and X − Y are independent random variables. This is a special case of the following important theorem. Let X and Y be independent random variables with a common distribution F. The random variables X + Y and X
7. Let X and Y be independent random variables with common probability density function f (x) = B e−x if x > 0 0 elsewhere. Find the joint probability density function of U = X + Y and V = eX.
6. Let X and Y be independent random variables with common probability density function f (x) =    1 x2 if x ≥ 1 0 elsewhere.Calculate the joint probability density function of U = X/Y and V = XY
5. Let −1/9 < c < 1/9 be a constant. Let p(x, y), the joint probability mass function of the random variables X and Y , be given by the following table: y x −1 0 1 −1 1/9 1/9 − c 1/9 + c 0 1/9 + c 1/9 1/9 − c 1 1/9 − c 1/9 + c 1/9 (a) Show that the probability mass function of X+Y is
4. From the interval (0, 1), two random numbers are selected independently. Show that the probability density function of their sum is given by g(t) =    t if 0 ≤ t < 1 2 − t if 1 ≤ t < 2 0 otherwise.
3. Let X ∼ N (0, 1) and Y ∼ N (0, 1) be independent random variables. Find the joint probability density function of R = √X2 + Y 2 and 5 = arctan(Y/X). Show that R and 5 are independent. Note that (R, 5) is the polar coordinate representation of (X, Y ).
2. Let X and Y be two positive independent continuous random variables with the probability density functions f1(x) and f2(y), respectively. Find the probability density function of U = X/Y. Hint: Let V = X; find the joint probability density function of U and V . Then calculate the marginal
1. Let X and Y be independent random numbers from the interval (0, 1). Find the joint probability density function of U = −2 ln X and V = −2 ln Y .
21. The lifetimes of batteries manufactured by a certain company are identically distributed with probability distribution and density functions F and f , respectively. Suppose that a battery manufactured by this company is installed at time 0 and begins to operate. If at time s an inspector finds
20. Let X and Y be discrete random variables with joint probability mass function p(x, y) = 1 e2y! (x − y)! , x = 0, 1, 2, . . . , y = 0, 1, 2, . . . , x, p(x, y) = 0, elsewhere. Find E(Y | X = x).
19. A point (X, Y ) is selected randomly from the triangle with vertices (0, 0), (0, 1), and (1, 0).(a) Find the joint probability density function of X and Y . (b) Calculate fX|Y (x|y). (c) Evaluate E(X | Y = y).
18. Let X and Y be continuous random variables with joint probability density function f (x, y) = B n(n − 1)(y − x)n−2 if 0 ≤ x ≤ y ≤ 1 0 otherwise. Find the conditional expectation of Y given that X = x.
17. A box contains 10 red and 12 blue chips. Suppose that 18 chips are drawn, one by one, at random and with replacement. If it is known that 10 of them are blue, show that the expected number of blue chips in the first nine draws is five.
16. Cards are drawn from an ordinary deck of 52, one at a time, randomly and with replacement. Let X and Y denote the number of draws until the first ace and the first king are drawn, respectively. Find E(X | Y = 5).
14. A point is selected at random and uniformly from the region R = $ (x, y): |x|+|y| ≤ 1 % . Find the conditional probability density function of X given Y = y. 15. Let $ N (t): t ≥ 0 % be a Poisson process. For s < t show that the conditional distribution of N (s) given N (t) = n is binomial
13. In a sequence of independent Bernoulli trials, let X be the number of successes in the first m trials and Y be the number of successes in the first n trials, m
12. Show that if $ N (t): t ≥ 0 % is a Poisson process, the conditional distribution of the first arrival time given N (t) = 1 is uniform on (0, t).
11. Leon leaves his office every day at a random time between 4:30 P.M. and 5:00 P.M. If he leaves t minutes past 4:30, the time it will take him to reach home is a random number between 20 and 20 + (2t)/3 minutes. Let Y be the number of minutes past 4:30 that Leon leaves his office tomorrow and X
10. The joint probability density function of X and Y is given by f (x, y) = B c e−x if x ≥ 0, |y| < x 0 otherwise.(a) Determine the constantc. (b) Find fX|Y (x|y) and fY |X(y|x). (c) Calculate E(Y | X = x) and Var(Y | X = x).
9. Let (X, Y ) be a random point from a unit disk centered at the origin. Find P (0 ≤ X ≤ 4/11 | Y = 4/5).
8. First a point Y is selected at random from the interval (0, 1). Then another point X is selected at random from the interval(Y, 1). Find the probability density function of X.
7. Let X and Y be continuous random variables with joint probability density function given by f (x, y) = B e−x(y+1) if x ≥ 0, 0 ≤ y ≤ e − 1 0 elsewhere. Calculate E(X | Y = y).
6. Let X and Y be continuous random variables with joint probability density function f (x, y) = B x + y if 0 ≤ x ≤ 1, 0 ≤ y ≤ 1 0 elsewhere. Calculate fX|Y (x|y).
5. Let X and Y be independent discrete random variables. Prove that for all y, E(X | Y = y) = E(X). Do the same for continuous random variables X and Y .
4. Let the conditional probability density function of X given that Y = y be given by fX|Y (x|y) = 3(x2 + y2) 3y2 + 1 , 0 < x < 1, 0 < y < 1. Find P (1/4 < X < 1/2 | Y = 3/4).
3. An unbiased coin is flipped until the sixth head is obtained. If the third head occurs on the fifth flip, what is the probability mass function of the number of flips?
2. Let the joint probability density function of continuous random variables X and Y be given by f (x, y) = B 2 if 0 < x < y < 1 0 elsewhere. Find fX|Y (x|y).
1. 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 pX|Y (x|y), P (X = 2 | Y = 1), and E(X | Y = 1).
26. Let f (x, y) be the joint probability density function of two continuous random variables; f is called circularly symmetrical if it is a function of G x2 + y2, the distance of (x, y) from the origin; that is, if there exists a function ϕ so that f (x, y) = ϕ( G x2 + y2 ). Prove that if X and
25. Suppose that X and Y are independent, identically distributed exponential random variables with mean 1/λ. Prove that X/(X + Y ) is uniform over (0, 1).
24. Let X and Y be two independent random points from the interval (0, 1). Calculate the probability distribution function and the probability density function of max(X, Y )/ min(X, Y ).
23. Let the joint probability density function of two random variables X and Y satisfy f (x, y) = g(x)h(y), −∞ < x < ∞, −∞ < y < ∞, where g and h are two functions from R to R. Show that X and Y are independent.
22. Let B and C be two independent random variables both having the following density function: f (x) =    3x2 26 if 1 < x < 3 0 otherwise. What is the probability that the quadratic equation X2 +BX +C = 0 has two real roots?
21. Let E be an event; the random variable IE, defined as follows, is called the indicator of E: IE = B 1 if E occurs 0 otherwise. Show that A and B are independent events if and only if IA and IB are independent random variables.
20. The lifetimes of mufflers manufactured by company A are random with the following density function: f (x) =    1 6 e−x/6 if x > 0 0 elsewhere.The lifetimes of mufflers manufactured by company B are random with the following density function: g(y) =  
19. Six brothers and sisters who are all either under 10 or in their early teens are having dinner with their parents and four grandparents. Their mother unintentionally feeds the entire family (including herself) a type of poisonous mushrooms that makes 20% of the adults and 30% of the children
18. A point is selected at random from the disk R = $ (x, y) ∈ R2 : x2 + y2 ≤ 1 % .Let X be the x-coordinate and Y be the y-coordinate of the point selected. Determine if X and Y are independent random variables.
17. Let X and Y be independent random points from the interval (0, 1). Find the probability density function of the random variable XY .
16. Let X and Y be independent random points from the interval (−1, 1). Find E 4 max(X, Y )5 .
15. Let X and Y be independent exponential random variables both with mean 1. Find E 4 max(X, Y )5 .
14. Let X and Y be two independent random variables with the same probability density function given by f (x) = B e−x if 0 < x < ∞ 0 elsewhere. Show that g, the probability density function of X/Y , is given by g(t) =    1 (1 + t)2 if 0 < t < ∞ 0 t ≤ 0.
13. Let the joint probability density function of X and Y be given by f (x, y) = B 2e−(x+2y) if x ≥ 0, y ≥ 0 0 otherwise. Find E(X2Y ).

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