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introduction to probability statistics

Introduction To Probability Models 12th Edition Sheldon M Ross - Solutions

96. Consider a large population of families, and suppose that the number of children in the different families are independent Poisson random variables with mean λ. Show that the number of siblings of a randomly chosen child is also Poisson distributed with mean λ.
95. For the left skip free random walk of Section 3.6.6 letI mage be the probability that the walk is never positive. Find β when .
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,where we use the convention thatI mage if eitherI mage orI mage. Now, consider a compound random variableI mage, where the
*93. Consider a sequence of independent trials, each of which is equally likely to result in any of the outcomesI mage. Say that a round begins with the first trial, and that a new round begins each time outcome 0 occurs. Let N denote the number of trials that it takes until all of the outcomes
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 pennies but picks up the other coins.(a) Find the expected amount of money that Josh picks up on his way
91. Find the expected number of flips of a coin, which comes up heads with probability p, that are necessary to obtain the pattern.
90. The number of accidents in each period is a Poisson random variable with mean 5. With , equal to the number of accidents in period n, find whena) Image;(b) .
89. LetI mage 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 determining defined by Justify the following formula:Image
88. In Section 3.6.3, we saw that if U is a random variable that is uniform onI mage and if, conditional onI mage is binomial with parameters n and p, then For another way of showing this result, let be independent uniform (0, 1) random variables. Define X by That is, if the variables are ordered
87. Recall that X is said to be a gamma random variable with parametersI mage if its density is(a) If Z is a standard normal random variable, show that Image is a gamma random variable with parameters Image (b) If Image are independent standard normal random variables, then Image is said to be a
*86. Each new book donated to a library must be processed. Suppose that the time it takes a librarian to process a book has mean 10 minutes and standard deviation 3 minutes. If a librarian has 40 books that must be processed one at a time,(a) approximate the probability that it will take more than
84. TeamsI mage are all scheduled to play each of the other teams 10 times. Whenever team i plays team j, team i is the winner with probabilityI mage, where(a) Approximate the probability that team 1 wins at least 20 games.Suppose now that we want to approximate the probability that team 2 wins at
83. WithI mage, show that Image
82. LetI mage denote the joint moment generating function ofI mage.(a) Explain how the moment generating function of Image, can be obtained from Image.(b) Show that Image are independent if and only if Image
81. Let X and Y be independent normal random variables, each having parameters μ andI mage. Show thatI mage is independent ofI mage.Hint: Find their joint moment generating function.
*80. Show that Image Hint: LetI mage be Poisson with mean n. Use the central limit theorem to show thatI mage.
79. If X is normally distributed with mean 1 and variance 4, use the tables to find .
78. LetI mage be independent Poisson random variables with mean 1.(a) Use the Markov inequality to get a bound on Image.(b) Use the central limit theorem to approximate Image.
77. Suppose that X is a random variable with mean 10 and variance 15. What can we say aboutI mage
76. Use Chebyshev's inequality to prove the weak law of large numbers. Namely, ifI mage are independent and identically distributed with mean μ and varianceI mage then, for anyI mage, Image
75. Consider Example 2.48. FindI mage in terms of theI mage.
*74. If X is Poisson with parameter λ, show that its Laplace transform is given by Image
73. Consider n people and suppose that each of them has a birthday that is equally likely to be any of the 365 days of the year.Furthermore, assume that their birthdays are independent, and let A be the event that no two of them share the same birthday. Define a“trial” for each of theI mage
72. Successive monthly sales are independent normal random variables with mean 100 and variance 100.(a) Find the probability that at least one of the next 5 months has sales above 115.(b) Find the probability that the total number of sales over the next 5 months exceeds 530.
*71. Show that the sum of independent identically distributed exponential random variables has a gamma distribution.
70. Calculate the moment generating function of a geometric random variable.
69. In deciding upon the appropriate premium to charge, insurance companies sometimes use the exponential principle, defined as follows. With X as the random amount that it will have to pay in claims, the premium charged by the insurance company is Image where a is some specified positive constant.
68. Let X and W be the working and subsequent repair times of a certain machine. LetI mage and suppose that the joint probability density of X and Y is(a) Find the density of X.(b) Find the density of Y.(c) Find the joint density of X and W.(d) Find the density of W.
67. Calculate the moment generating function of the uniform distribution on I(mage). Obtain andI mage by differentiating.
*66. Show that the random variablesI mage are independent if for each ImageI mage is independent ofI mage.Hint: Image are independent if for any setsI mage Image On the other handI mage is independent ofI mage if for any setsI mage Image
65. The number of traffic accidents on successive days are independent Poisson random variables with mean 2.(a) Find the probability that 3 of the next 5 days have two accidents.(b) Find the probability that there are a total of six accidents over the next 2 days.(c) If each accident is
*64. Show that when X is the number of men who select their own hats in Example 2.30.
63. Let X denote the number of white balls selected when k balls are chosen at random from an urn containing n white and m black balls.(a) Compute Image.(b) Let, for Image,Compute in two ways by expressing X first as a function of the Image and then of theI mage.
62. LetI mage denote a set of n numbers, and consider any permutation of these numbers. We say that there is an inversion ofI mage and Image in the permutation ifI mage andI mage precedesI mage. For instance the permutation 4, 2, 1, 5, 3 has 5 inversions—(4, 2), (4, 1), (4, 3), (2, 1), (5, 3).
61. LetI mage be a sequence of independent identically distributed continuous random variables. We say that a record occurs at time n ifI mage. That is,I mage is a record if it is larger than each of. Show(a) Image;(b) Image;(c) Image;(d) Let Image. Show Image.Hint: For (b) and (c) represent the
60. Let X and Y be independent random variables with meansI mage andI mage and variancesI mage andI mage. Show that
59. Let , andI mage be independent continuous random variables with a common distribution function F and let(a) Argue that the value of p is the same for all continuous distribution functions F.(b) Find p by integrating the joint density function over the appropriate region.(c) Find p by using the
58. An urn contains 2n balls, of which r are red. The balls are randomly removed in n successive pairs. Let X denote the number of pairs in which both balls are red.(a) Find .(b) Find Image.
57. Suppose that X and Y are independent binomial random variables with parameters (Image) and (Image). Argue probabilistically (no computations necessary) thatI mage is binomial with parameters (Image).
56. There are n types of coupons. Each newly obtained coupon is, independently, type i with probabilityI mage. Find the expected number and the variance of the number of distinct types obtained in a collection of k coupons.
55. Suppose that the joint probability mass function of X and Y is Image(a) Find the probability mass function of Y.(b) Find the probability mass function of X.(c) Find the probability mass function of Image.
54. Each member of a population is either type 1 with probability Image or type 2 with probabilityI mage. Independent of other pairs, two individuals of the same type will be friends with probability α, whereas two individuals of different types will be friends with probability β. LetI mage be
53. If X is uniform over (0, 1), calculateI mage and VarI(mage).
52.(a) Calculate for the maximum random variable of Exercise 37.(b) Calculate for X as in Exercise 33.(c) Calculate for X as in Exercise 34.
51. A coin, having probability p of landing heads, is flipped until a head appears for the rth time. Let N denote the number of flips required. Calculate .Hint: There is an easy way of doing this. It involves writing N as the sum of r geometric random variables.
50. Let c be a constant. Show that(a) Image;(b) Image.
*49. Prove thatI mage. When do we have equality?
48. For any event A, we define the random variableI mage, called the indicator variable for A, by letting it equal 1 when A occurs and 0 when A does not. Now, ifI mage is a nonnegative random variable for allI mage, then it follows from a result in real analysis called Fubini's theorem that Suppose
*47. Consider three trials, each of which is either a success or not. Let X denote the number of successes. Suppose thatI mage.(a) What is the largest possible value of Image?(b) What is the smallest possible value of Image?In both cases, construct a probability scenario that results inI mage
46. If X is a nonnegative integer valued random variable, show that Define the sequence of random variables Image, by Image Now express X in terms of the .(b) If X and Y are both nonnegative integer valued random variables, show that
45. A total of r keys are to be put, one at a time, in k boxes, with each key independently being put in box i with probabilityI mage. Each time a key is put in a nonempty box, we say that a collision occurs.Find the expected number of collisions.
44. In Exercise 43, let Y denote the number of red balls chosen after the first but before the second black ball has been chosen.(a) Express Y as the sum of n random variables, each of which is equal to either 0 or 1.(b) Find Image.(c) Compare Image to obtained in Exercise 43.(d) Can you explain
43. An urn containsI mage balls, of which n are red and m are black.They are withdrawn from the urn, one at a time and without replacement. Let X be the number of red balls removed before the first black ball is chosen. We are interested in determining . To obtain this quantity, number the red
42. Suppose that each coupon obtained is, independent of what has been previously obtained, equally likely to be any of m different types. Find the expected number of coupons one needs to obtain in order to have at least one of each type.Hint: Let X be the number needed. It is useful to represent X
41. Consider the case of arbitrary p in Exercise 29. Compute the expected number of changeovers.
40. Suppose that two teams are playing a series of games, each of which is independently won by team A with probability p and by team B with probability . The winner of the series is the first team to win four games. Find the expected number of games that are played, and evaluate this quantity
39. An urn has 8 red and 12 blue balls. Suppose that balls are chosen at random and removed from the urn, with the process stopping when all the red balls have been removed. Let X be the number of balls that have been removed when the process stops.(a) Find Image.(b) Find the probability that a
38. LetI mage be independent and identically distributed continuous random variables with distribution function F, and meanI mage. Let Image be the values arranged in increasing order. That is, forI mage, Image is the ith smallest ofI mage.(a) Find Image.(b) Let Image. What is the distribution of
37. LetI mage be independent random variables, each having a uniform distribution over . Let M= maximumI mage. Show that the distribution function ofI mage, is given by Image What is the probability density function of M?
36. A point is uniformly distributed within the disk of radius 1. That is, its density is Find the probability that its distance from the origin is less than .
35. The density of X is given by Image What is the distribution function of X? FindI mage.
34. Let the probability density of X be given by(a) What is the value of c?(b) Image
33. Let X be a random variable with probability density(a) What is the value of c?(b) What is the cumulative distribution function of X?
32. If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize isI mage, what is the (approximate)probability that you will win a prize (a) at least once, (b) exactly once, (c) at least twice?
31. Compare the Poisson approximation with the correct binomial probability for the following cases:(a) Image when Image.(b) Image.(c) Image.(d) Image.
30. Let X be a Poisson random variable with parameter λ. Show that Image increases monotonically and then decreases monotonically as i increases, reaching its maximum when i is the largest integer not exceeding λ.Hint: ConsiderI mage.
29. Consider n independent flips of a coin having probability p of landing heads. Say a changeover occurs whenever an outcome differs from the one preceding it. For instance, if the results of the flips areI mage, then there are a total of five changeovers. IfI mage, what is the probability there
28. Suppose that we want to generate a random variable X that is equally likely to be either 0 or 1, and that all we have at our disposal is a biased coin that, when flipped, lands on heads with some (unknown) probability p. Consider the following procedure:1. Flip the coin, and let 01, either
*27. A fair coin is independently flipped n times, k times by A and Image times by B. Show that the probability that A and B flip the same number of heads is equal to the probability that there are a total of k heads.
26. Find the expected number of games that are played when(a) Image;(b) .In both cases, show that this number is maximized whenI mage.
25. IfI mage, find the probability that a total of 7 games are played.Also show that this probability is maximized whenI mage.
24. The probability mass function of X is given by Give a possible interpretation of the random variable X.Hint: See Exercise 23.In Exercises 25 and 26, suppose that two teams are playing a series of games, each of which is independently won by team A with probability p and by team B with
*23. A coin having probability p of coming up heads is successively flipped until the rth head appears. Argue that X, the number of flips required, will beI mage, with probability This is known as the negative binomial distribution.Hint: How many successes must there be in the first trials?
22. If a fair coin is successively flipped, find the probability that a head first appears on the fifth trial.
21. LetI mage andI mage be independent binomial random variables, withI mage having parameters ,I mage.(a) Find Image.(b) Find Image.(c) Find Image.
20. In this problem we employ the multinomial distribution to solve an extension of the birthday problem. Assuming that each of n individuals is, independently of others, equally likely to have their birthday be any of the 365 days of the year, we want to derive an expression for the probability
19. In Exercise 17, letI mage denote the number of times the ith outcome appears,I mage. What is the probability mass function of Image?
18. In Exercise 17, letI mage denote the number of times that the ith type outcome occurs,I mage(a) For Image, use the definition of conditional probability to find Image(b) What can you conclude about the conditional distribution of Image given that Image?(c) Give an intuitive explanation for your
17. Suppose that an experiment can result in one of r possible outcomes, the ith outcome having probabilityI mage,I mage. If n of these experiments are performed, and if the outcome of any one of the n does not affect the outcome of the other experiments, then show that the probability that the
*16. An airline knows that 5 percent of the people making reservations on a certain flight will not show up. Consequently, their policy is to sell 52 tickets for a flight that can hold only 50 passengers. What is the probability that there will be a seat available for every passenger who shows up?
15. Let X be binomially distributed with parameters n and p. Show that as k goes from 0 toI mage increases monotonically, then decreases monotonically, reaching its largest value(a) in the case that Image is an integer, when k equals either or Image,(b) in the case that Image is not an integer,
14. Suppose X has a binomial distribution with parameters 6 andI mage. Show thatI mage is the most likely outcome.
13. An individual claims to have extrasensory perception (ESP). As a test, a fair coin is flipped ten times, and he is asked to predict in advance the outcome. Our individual gets seven out of ten correct.What is the probability he would have done at least this well if he had no ESP? (Explain why
12. On a multiple-choice exam with three possible answers for each of the five questions, what is the probability that a student would get four or more correct answers just by guessing?
*11. A ball is drawn from an urn containing three white and three black balls. After the ball is drawn, it is then replaced and another ball is drawn. This goes on indefinitely. What is the probability that of the first four balls drawn, exactly two are white?
10. Suppose three fair dice are rolled. What is the probability at most one six appears?
9. If the distribution function of F is given by Image calculate the probability mass function of X.
8. Suppose the distribution function of X is given by Image What is the probability mass function of X?
7. Suppose a coin having probability 0.7 of coming up heads is tossed three times. Let X denote the number of heads that appear in the three tosses. Determine the probability mass function of X.
6. Suppose five fair coins are tossed. Let E be the event that all coins land heads. Define the random variable For what outcomes in the original sample space does equal 1? What isI mage?
5. If the die in Exercise 4 is assumed fair, calculate the probabilities associated with the random variables in (a)–(d).
*4. Suppose a die is rolled twice. What are the possible values that the following random variables can take on?(a) The maximum value to appear in the two rolls.(b) The minimum value to appear in the two rolls.(c) The sum of the two rolls.(d) The value of the first roll minus the value of the
3. In Exercise 2, if the coin is assumed fair, then, for , what are the probabilities associated with the values that X can take on?
2. Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed n times. What are the possible values of X?
1. An urn contains five red, three orange, and two blue balls. Two balls are randomly selected. What is the sample space of this experiment? Let X represent the number of orange balls selected.What are the possible values of X? CalculateI mage.
51. There is a 40 percent chance of rain on Monday; a 30 percent chance of rain on Tuesday; and a 20 percent chance of rain on both days. It did not rain on Monday. What is the probability it will rain on Tuesday.
50. If is a sequence of events then is defined as the set of points that are in an infinite number of the events ;and is defined as the set of points that are in all but a finite number of the events .(a) If is an increasing sequence of events, show that(b) If is a decreasing sequence of events,
49. Prove Proposition 1.1 for a sequence of decreasing events.
47. For a fixed event B, show that the collection , defined for all events A, satisfies the three conditions for a probability.Conclude from this that Then directly verify the preceding equation.*48. Sixty percent of the families in a certain community own their own car, thirty percent own their
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 of his fellow prisoners will be set free, claiming that there would be no harm in divulging this

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