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theory of probability

Questions and Answers of Theory Of Probability

52. Suppose that X and Y are independent random variables with probability density functions fX and fY . Determine a one-dimensional integral expression for P{X +Y < x}.
53. Suppose X is a Poisson random variable with mean λ. The parameter λ is itself a random variable whose distribution is exponential with mean 1. Show that P{X = n} = ( 1 2 )n+1.
54. Independent trials, each resulting in a success with probability p, are performed until k consecutive successful trials have occurred. Let X be the total number of successes in these trial, and
55. In the preceding problem let Mk = E[X]. Derive a recursive equation for Mk and then solve.Hint: Start by writing Xk = Xk−1 + Ak−1,k , where Xi is the total number of successes attained up to
56. Data indicate that the number of traffic accidents in Berkeley on a rainy day is a Poisson random variable with mean 9, whereas on a dry day it is a Poisson random variable with mean 3. Let X
57. The number of storms in the upcoming rainy season is Poisson distributed but with a parameter value that is uniformly distributed over (0, 5). That is, is uniformly distributed over (0, 5), and
58. Suppose that the conditional distribution of N, given that Y = y, is Poisson with mean y. Further suppose that Y is a gamma random variable with parameters(r, λ), where r is a positive integer.
59. Suppose each new coupon collected is, independent of the past, a type i coupon with probability pi . A total of n coupons is to be collected. Let Ai be the event that there is at least one type i
60. Two players alternate flipping a coin that comes up heads with probability p. The first one to obtain a head is declared the winner.We are interested in the probability that the first player to
61. Suppose in Exercise 29 that the shooting ends when the target has been hit twice.Let mi denote the mean number of shots needed for the first hit when player i?
62. A, B, and C are evenly matched tennis players. Initially A and B play a set, and the winner then plays C. This continues, with the winner always playing the waiting player, until one of the
63. Suppose there are n types of coupons, and that the type of each new coupon obtained is independent of past selections and is equally likely to be any of the n types. Suppose one continues
64. A and B roll a pair of dice in turn, with A rolling first. A’s objective is to obtain a sum of 6, and B’s is to obtain a sum of 7. The game ends when either player reaches his or her
65. The number of red balls in an urn that contains n balls is a random variable that is equally likely to be any of the values 0, 1,..., n. That is, P{i red, n − i non-red} =1 n + 1, i = 0,..., n
66. The opponents of soccer team A are of two types: either they are a class 1 or a class 2 team. The number of goals team A scores against a class i opponent is a Poisson random variable with mean
67. A coin having probability p of coming up heads is continually flipped. Let Pj(n)denote the probability that a run of j successive heads occurs within the first n flips.(a) Argue that Pj(n) = Pj(n
68. In a knockout tennis tournament of 2n contestants, the players are paired and play a match. The losers depart, the remaining 2n−1 players are paired, and they play a match. This continues for n
69. In the match problem, say that (i, j),i < j, is a pair if i chooses j’s hat and j chooses i’s hat.(a) Find the expected number of pairs.
70. Let N denote the number of cycles that result in the match problem.(a) Let Mn = E[N], and derive an equation for Mn in terms of M1,..., Mn−1.(b) Let Cj denote the size of the cycle that
71. Use Equation (3.13) to obtain Equation (3.9).Hint: First multiply both sides of Equation (3.13) by n, then write a new equation by replacing n with n − 1, and then subtract the former from the
72. In Example 3.28 show that the conditional distribution of N given that U1 = y is the same as the conditional distribution of M given that U1 = 1 − y. Also, show that E[N|U1 = y] = E[M|U1 = 1
73. Suppose that we continually roll a die until the sum of all throws exceeds 100.What is the most likely value of this total when you stop?
74. There are five components. The components act independently, with component i working with probability pi,i = 1, 2, 3, 4, 5. These components form a system as shown in Figure 3.7.The system is
75. This problem will present another proof of the ballot problem of Example 3.27.(a) Argue that Pn,m = 1 − P{A and B are tied at some point}(b) Explain why
76. Consider a gambler who on each bet either wins 1 with probability 18/38 or loses 1 with probability 20/38. (These are the probabilities if the bet is that a roulette wheel will land on a
77. Show that(a) E[XY |Y = y] = yE[X|Y = y](b) E[g(X, Y )|Y = y] = E[g(X, y)|Y = y](c) E[XY ] = E[Y E[X|Y ]]
78. In the ballot problem (Example 3.27), compute P {A is never behind}.
79. An urn contains n white and m black balls that are removed one at a time. If n > m, show that the probability that there are always more white than black balls in the urn (until, of course, the
80. A coin that comes up heads with probability p is flipped n consecutive times.What is the probability that starting with the first flip there are always more heads than tails that have appeared?
81. Let Xi,i 1, be independent uniform (0, 1) random variables, and define N by N = min{n: Xn < Xn−1}where X0 = x. Let f (x) = E[N].
82. Let X1, X2,... be independent continuous random variables with a common distribution function F and density f = F, and for k 1 let Nk = min{n k: Xn = kth largest of X1,..., Xn}(a) Show that
83. An urn contains n balls, with ball i having weight wi,i = 1,..., n. The balls are withdrawn from the urn one at a time according to the following scheme: When S is the set of balls that remains,
84. Suppose in Example 3.32 that a point is only won if the winner of the rally was the server of that rally.(a) If A is currently serving, what is the probability that A wins the next point?(b)
85. In the list problem, when the Pi are known, show that the best ordering (best in the sense of minimizing the expected position of the element requested) is to place the elements in decreasing
86. Consider the random graph of Section 3.6.2 when n = 5. Compute the probability distribution of the number of components and verify your solution by using it to compute E[C] and then comparing
87. (a) From the results of Section 3.6.3 we can conclude that there are n+m−1 m−1nonnegative integer valued solutions of the equation x1 +··· + xm = n.Prove this directly.(b) How many
88. In Section 3.6.3, we saw that if U is a random variable that is uniform on (0, 1)and if, conditional on U = p, X is binomial with parameters n and p, then P{X = i} =1 n + 1, i = 0, 1,..., n For
89. Let I1,..., In be independent random variables, each of which is equally likely to be either 0 or 1. A well-known nonparametric statistical test (called the signed rank test) is concerned with
90. The number of accidents in each period is a Poisson random variable with mean 5. With Xn, n 1, equal to the number of accidents in period n, find E[N] when(a) N = min (n: Xn−2 = 2, Xn−1 =
91. Find the expected number of flips of a coin, which comes up heads with probability p, that are necessary to obtain the pattern h, t, h, h, t, h, t, h.
92. The number of coins that Josh spots when walking to work is a Poisson random variable with mean 6. Each coin is equally likely to be a penny, a nickel, a dime, or a quarter. Josh ignores the
93. Consider a sequence of independent trials, each of which is equally likely to result in any of the outcomes 0, 1,..., m. Say that a round begins with the first trial, and that a new round begins
94. Let N be a hypergeometric random variable having the distribution of the number of white balls in a random sample of size r from a set of w white and b blue balls.That is, P{N = n} =w n b
95. For the left skip free random walk of Section 3.6.6 let β = P(Sn 0 for all n)be the probability that the walk is never positive. Find β when E[Xi] < 0.
96. Consider a large population of families, and suppose that the number of children in the different families are independent Poisson random variables with meanλ. Show that the number of siblings
97. Use the conditional variance formula to find the variance of a geometric random variable.
98. For a compound random variable S = N i=1 Xi , find Cov(N, S).
99. Let N be the number of trials until k consecutive successes have occurred, when each trial is independently a success with probability p.(a) What is P(N = k)?(b) Argue that
1. A box contains three marbles: one red, one green, and one blue. Consider an experiment that consists of taking one marble from the box then replacing it in the box and drawing a second marble from
2. Repeat Exercise 1 when the second marble is drawn without replacing the first marble.
3. A coin is to be tossed until a head appears twice in a row. What is the sample space for this experiment? If the coin is fair, what is the probability that it will be tossed exactly four times?
4. Let E, F, G be three events. Find expressions for the events that of E, F, G(a) only F occurs,(b) both E and F but not G occur,(c) at least one event occurs,(d) at least two events occur,(e) all
5. An individual uses the following gambling system at Las Vegas. He bets $1 that the roulette wheel will come up red. If he wins, he quits. If he loses then he makes the same bet a second time only
6. Show that E(FUG) = EF U EG.
7. Show that (EUF)c = EcFc.
8. If P(E) = 0.9 and P(F) = 0.8, show that P(EF) ≥ 0.7. In general, show that:P(EF) > P(E) + P(F) - 1 This is known as Bonferroni's inequality.
9. We say that E ⊆ F if every point in E is also in F. Show that if E ⊆ F, then:P(F) = P(E) + P(FEc) > P(E)
10. Show that$$P(\bigcup_{i=1}^{n} E_i) ≤ \sum_{i=1}^{n} P(E_i)$$This is known as Boole's inequality.Hint: Either use Equation (1.2) and mathematical induction, or else show that$\bigcup_{i=1}^{n}
11. If two fair dice are tossed, what is the probability that the sum is i, i =2, 3, ..., 12?
12. Let E and F be mutually exclusive events in the sample space of an experiment. Suppose that the experiment is repeated until either event E or event F occurs. What does the sample space of this
13. The dice game craps is played as follows. The player throws two dice, and if the sum is seven or eleven, then she wins. If the sum is two, three, or twelve, then she loses. If the sum is anything
14. The probability of winning on a single toss of the dice is p. A starts, and if he fails, he passes the dice to B, who then attempts to win on her toss. They continue tossing the dice back and
15. Argue that E = EFU EFC, EUF = EU FEC.
16. Use Exercise 15 to show that P(EUF) = P(E) + P(F) – P(EF).
17. Suppose each of three persons tosses a coin. If the outcome of one of the tosses differs from the other outcomes, then the game ends. If not, then the persons start over and retoss their coins.
18. Assume that each child who is born is equally likely to be a boy or a girl.If a family has two children, what is the probability that both are girls given that(a) the eldest is a girl, (b) at
19. Two dice are rolled. What is the probability that at least one is a six? If the two faces are different, what is the probability that at least one is a six?
20. Three dice are thrown. What is the probability the same number appears on exactly two of the three dice?
21. Suppose that 5 percent of men and 0.25 percent of women are color-blind.A color-blind person is chosen at random. What is the probability of this person being male? Assume that there are an equal
22. A and B play until one has 2 more points than the other. Assuming that each point is independently won by A with probability p, what is the probability they will play a total of 2n points? What
23. For events E1, E2, ..., En show that P(E1 E2 ... En) = P(E1)P(E2|E1)P(E3|E1E2) ... P(En|E1... En-1)
24. In an election, candidate A receives n votes and candidate B receives m votes, where n > m. Assume that in the count of the votes all possible orderings of the n + m votes are equally likely. Let
25. Two cards are randomly selected from a deck of 52 playing cards.(a) What is the probability they constitute a pair (that is, that they are of the same denomination)?(b) What is the conditional
26. A deck of 52 playing cards, containing all 4 aces, is randomly divided into 4 piles of 13 cards each. Define events E1, E2, E3, and E4 as follows::1 E₁ = {the first pile has exactly 1 ace}, E2
27. Suppose in Exercise 26 we had defined the events E;, i = 1, 2, 3, 4, by E₁ = {one of the piles contains the ace of spades}, E2 = {the ace of spades and the ace of hearts are in different
28. If the occurrence of B makes A more likely, does the occurrence of A make B more likely?
29. Suppose that P(E) = 0.6. What can you say about P(E/F) when(a) E and F are mutually exclusive?(b) EC F?(c) FCE?
30. Bill and George go target shooting together. Both shoot at a target at the same time. Suppose Bill hits the target with probability 0.7, whereas George, independently, hits the target with
31. What is the conditional probability that the first die is six given that the sum of the dice is seven?
32. Suppose all *n* men at a party throw their hats in the center of the room.Each man then randomly selects a hat. Show that the probability that none of the*n* men selects his own hat
33. In a class there are four freshman boys, six freshman girls, and six sophomore boys. How many sophomore girls must be present if sex and class are to be independent when a student is selected at
34. Mr. Jones has devised a gambling system for winning at roulette. When he bets, he bets on red, and places a bet only when the ten previous spins of the roulette have landed on a black number. He
35. A fair coin is continually flipped. What is the probability that the first four flips are(a) H, H, H, H?(b) Τ, Η, Η, H?(c) What is the probability that the pattern Τ, Η, Η, H occurs before
36. Consider two boxes, one containing one black and one white marble, the other, two black and one white marble. A box is selected at random and a marble is drawn at random from the selected box.
37. In Exercise 36, what is the probability that the first box was the one selected given that the marble is white?
38. Urn 1 contains two white balls and one black ball, while urn 2 contains one white ball and five black balls. One ball is drawn at random from urn 1 and placed in urn 2. A ball is then drawn from
39. Stores A, B, and C have 50, 75, and 100 employees, and, respectively, 50, 60, and 70 percent of these are women. Resignations are equally likely among all employees, regardless of sex. One
40. (a) A gambler has in his pocket a fair coin and a two-headed coin. He selects one of the coins at random, and when he flips it, it shows heads. What is the probability that it is the fair coin?
41. In a certain species of rats, black dominates over brown. Suppose that a black rat with two black parents has a brown sibling.(a) What is the probability that this rat is a pure black rat (as
42. There are three coins in a box. One is a two-headed coin, another is a fair coin, and the third is a biased coin that comes up heads 75 percent of the time.When one of the three coins is selected
43. Suppose we have ten coins which are such that if the *i*th one is flipped then heads will appear with probability *i*/10, *i* = 1,2,..., 10. When one of the coins is randomly selected and
44. Urn 1 has five white and seven black balls. Urn 2 has three white and twelve black balls. We flip a fair coin. If the outcome is heads, then a ball from urn 1 is selected, while if the outcome is
45. An urn contains *b* black balls and *r* red balls. One of the balls is drawn at random, but when it is put back, in the urn *c* additional balls of the same color are put in with it. Now suppose
46. Three prisoners are informed by their jailer that one of them has been chosen at random to be executed, and the other two are to be freed. Prisoner A asks the jailer to tell him privately which
47. For a fixed event *B*, show that the collection *P(A|B)*, defined for all events *A*, satisfies the three conditions for a probability. Conclude from this that$$P(A|B) = P(ABC)P(C|B) +
2. Give an approach for simulating a random variable having probability density function f(x)=30(x-2x + x) 0
1. The random variable X has probability density function f(x) = Ce (a) Find the value of the constant C. 0
15. (a) Verify that the minimum of (4.1) occurs when a is as given by (4.2).(b) Verify that the minimum of (4.1) is given by (4.3).16. Let X be a random variable on (0, 1) whose density is f(x). Show
14. In Example 4a we have shown that$$E[(1 - √V)²] = E[(1 - √U)²] = \frac{4}{\pi}$$when V is uniform (-1, 1) and U is uniform (0, 1). Show that$$Var[(1 - √V)²] = Var[(1 - √U)²]$$and find
13. Let (X, Y) be uniformly distributed in the circle of radius 1 centered at the origin. Its joint density is thus$$f(x,y)= \frac{1}{\pi}\qquad 0 \le x^2 + y^2 \le 1$$Let R = (X² + Y²)¹/² and θ

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