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theory of probability

Questions and Answers of Theory Of Probability

61. Let X1, X2,... be a sequence of independent identically distributed continuous random variables. We say that a record occurs at time n if Xn > max (X1,..., Xn−1). That is, Xn is a record if it
60. Let X and Y be independent random variables with means μx and μy and variances σ2 x and σ2 y . Show that Var(XY ) = σ2 x σ2 y + μ2 yσ2 x + μ2 xσ2 y
59. Let X1, X2, X3, and X4 be independent continuous random variables with a common distribution function F and let p = P{X1 < X2 > X3 < X4}(a) Argue that the value of p is the same for all
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 E[X].(b) Find Var(X).
57. Suppose that X and Y are independent binomial random variables with parameters (n, p) and (m, p). Argue probabilistically (no computations necessary) that X + Y is binomial with parameters (n +
56. There are n types of coupons. Each newly obtained coupon is, independently, type i with probability pi,i = 1,..., n. Find the expected number and the variance of the number of distinct types
55. Suppose that the joint probability mass function of X and Y is P(X = i, Y = j) = j ie−2λλj/ j!, 0 ≤ i ≤ j(a) Find the probability mass function of Y .(b) Find the probability mass
54. Let X and Y each take on either the value 1 or −1. Let
53. If X is uniform over (0, 1), calculate E[Xn] and Var(Xn).
52. (a) Calculate E[X] for the maximum random variable of Exercise 37.(b) Calculate E[X] for X as in Exercise 33.(c) Calculate E[X] 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 E[N].Hint: There is an easy way of doing
50. Let c be a constant. Show that(a) Var(cX) = c2Var(X);(b) Var(c + X) = Var(X).
49. Prove that E[X2] ≥ (E[X])2. When do we have equality?
48. If X is a nonnegative random variable, and g is a differential function with g(0) = 0, then E[g(X)] = ∞0 P(X > t)g(t)dt Prove the preceding when X is a continuous random variable.
47. Consider three trials, each of which is either a success or not. Let X denote the number of successes. Suppose that E[X] = 1.8.(a) What is the largest possible value of P{X = 3}?(b) What is the
46. If X is a nonnegative integer valued random variable, show that(a) E[X] = ∞n=1 P{X ≥ n} = ∞n=0 P{X > n}
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 probability pi, k i=1 pi = 1. Each time a key is put in a nonempty box, we say that
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
43. An urn contains n + m 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
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
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 1 − p.The winner of the series is the first
39. The random variable X has the following probability mass function:p(1) = 1 2 , p(2) = 1 3 , p(24) = 1 6Calculate E[X].
38. If the density function of X equals f (x) =ce−2x , 0 < x < ∞0, x < 0 findc. What is P{X > 2}?
37. Let X1, X2,..., Xn be independent random variables, each having a uniform distribution over (0,1). Let M = maximum (X1, X2,..., Xn). Show that the distribution function of M, FM (·), is given by
36. A point is uniformly distributed within the disk of radius 1. That is, its density is f (x, y) = C, 0 ≤ x2 + y2 ≤ 1
35. The density of X is given by f (x) =10/x2, for x > 10 0, for x ≤ 10 What is the distribution of X? Find P{X > 20}.
34. Let the probability density of X be given by f (x) =c(4x − 2x2), 0 < x < 2 0, otherwise(a) What is the value of c?(b) P - 1 2 < X < 3 2.=?
32. If you buy a lottery ticket in 50 lotteries, in each of which your chance of winning a prize is 1 100 , what is the (approximate) probability that you will win a prize (a)at least once, (b)
31. Compare the Poisson approximation with the correct binomial probability for the following cases:(a) P{X = 2} when n = 8, p = 0.1.(b) P{X = 9} when n = 10, p = 0.95.(c) P{X = 0} when n = 10, p =
30. Let X be a Poisson random variable with parameter λ. Show that P{X = i}increases monotonically and then decreases monotonically as i increases, reaching its maximum when i is the largest integer
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
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
27. A fair coin is independently flipped n times, k times by A and n − k 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
26. Find the expected number of games that are played when(a) i = 2;(b) i = 3.
25. If i = 4, find the probability that a total of 7 games are played. Also show that this probability is maximized when p = 1/2.
24. The probability mass function of X is given by p(k) =r + k − 1 r − 1pr(1 − p)k , k = 0, 1,...Give a possible interpretation of the random variable X.Hint: See Exercise 23.In Exercises 25
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 be n, n ≥ r, with probability P{X = n} = n
22. If a fair coin is successively flipped, find the probability that a head first appears on the fifth trial.
21. In Exercise 20, what is the probability that our store owner sells three or more televisions on that day?
20. A television store owner figures that 50 percent of the customers entering his store will purchase an low end television set, 20 percent will purchase a high end television set, and 30 percent
19. In Exercise 17, let Xi denote the number of times the ith outcome appears, i =1,...,r. What is the probability mass function of X1 + X2 + ... + Xk ?
18. In Exercise 17, let Xi denote the number of times that the ith type outcome occurs, i = 1,...,r.(a) For 0 ≤ j ≤ n, use the definition of conditional probability to find P(Xi =xi,i = 1,...,r
17. Suppose that an experiment can result in one of r possible outcomes, the ith outcome having probability pi , i = 1,...,r, r i=1 pi = 1. If n of these experiments are performed, and if the outcome
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
15. Let X be binomially distributed with parameters n and p. Show that as k goes from 0 to n, P(X = k) increases monotonically, then decreases monotonically reaching its largest value(a) in the case
14. Suppose X has a binomial distribution with parameters 6 and 1 2 . Show that X = 3 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
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
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 F(b) =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩0, b < 0 12 , 0 ≤ b < 1 35 , 1 ≤ b < 2 45 , 2 ≤ b < 3 910 , 3 ≤ b < 3.5
8. Suppose the distribution function of X is given by F(b) =⎧⎪⎨⎪⎩0, b < 0 12 , 0 ≤ b < 1 1, 1 ≤ b < ∞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 IE IE =1, if E occurs 0, if Ec occurs For what outcomes in the original sample space
5. If the die in Exercise 4 is assumed fair, calculate the probabilities associated with the random variables in (i)–(iv).
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
3. In Exercise 2, if the coin is assumed fair, then, for n = 2, 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
48. Sixty percent of the families in a certain community own their own car, thirty percent own their own home, and twenty percent own both their own car and their own home. If a family is randomly
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(A|BC)P(C|B) +
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
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 that we
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
43. The blue-eyed gene for eye color is recessive, meaning that both the eye genes of an individual must be blue for that individual to be blue eyed. Jo (F) and Joe (M)are both brown-eyed individuals
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
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
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
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
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
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.
35. A fair coin is continually flipped. What is the probability that the first four flips are(a) H, H, H, H?(b) T, H, H, H?(c) What is the probability that the pattern T, H, H, H occurs before the
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
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
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 is 12!− 1
31. What is the conditional probability that the first die is six given that the sum of the dice is seven?
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
29. Suppose that P(E) = 0.6. What can you say about P(E|F) when(a) E and F are mutually exclusive?(b) E ⊂ F?(c) F ⊂ E?
28. If the occurrence of B makes A more likely, does the occurrence of A make B more likely?
27. Suppose in Exercise 26 we had defined the events Ei,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 piles},
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:E1 = {the first pile has exactly 1 ace}, E2 =
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
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
23. For events E1, E2,..., En show that?
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
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
20. Three dice are thrown.What is the probability the same number appears on exactly two of the three dice?
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
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.
16. Use Exercise 15 to show that P(E ∪ F) = P(E) + P(F) − P(EF).
15. Argue that E = EF ∪ EFc, E ∪ F = E ∪ FEc.
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
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
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 new
11. If two fair dice are tossed, what is the probability that the sum isi,i = 2, 3,..., 12?
10. Show that Pn i=1 Ein i=1 P(Ei)This is known as Boole’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)
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.
7. Show that (E ∪ F)c = EcFc.

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