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theory of probability

Questions and Answers of Theory Of Probability

Suppose that families migrate to an area at a Poisson rate λ = 2 per week. If the number of people in each family is independent and takes on the values 1, 2, 3, 4 with respective probabilities 16,
Siegbert runs a hot dog stand that opens at 8 A.M. From 8 until 11 A.M. customers seem to arrive, on the average, at a steadily increasing rate that starts with an initial rate of 5 customers per
Suppose that in 100 units of operating time 20 bugs are discovered of which two resulted in exactly one, and three resulted in exactly two, errors. Then we would estimate that (100) is something
There are m different types of coupons. Each time a person collects a coupon it is, independently of ones previously obtained, a type j coupon with probability pj,mj=1 pj = 1. Let N denote the
Consider a system in which individuals at any time are classified as being in one of r possible states, and assume that an individual changes states in accordance with a Markov chain having
Suppose nonnegative offers to buy an item that you want to sell arrive according to a Poisson process with rate λ. Assume that each offer is the value of a continuous random variable having density
If immigrants to area A arrive at a Poisson rate of ten per week, and if each immigrant is of English descent with probability 1 12 , then what is the probability that no people of English descent
Suppose that people immigrate into a territory at a Poisson rateλ = 1 per day.(a) What is the expected time until the tenth immigrant arrives?(b) What is the probability that the elapsed time
Suppose that customers are in line to receive service that is provided sequentially by a server; whenever a service is completed, the next person in line enters the service facility. However, each
There are n cells in the body, of which cells 1, . . . , k are target cells.Associated with each cell is a weight, with wi being the weight associated with cell i, i = 1, . . . , n. The cells are
Suppose you arrive at a post office having two clerks at a moment when both are busy but there is no one else waiting in line. You will enter service when either clerk becomes free. If service times
(Analyzing Greedy Algorithms for the Assignment Problem)A group of n people is to be assigned to a set of n jobs, with one person assigned to each job. For a given set of n2 values Cij, i, j = 1, . .
A store must decide how much of a certain commodity to order so as to meet next month’s demand, where that demand is assumed to have an exponential distribution with rate λ. If the commodity costs
The dollar amount of damage involved in an automobile accident is an exponential random variable with mean 1000. Of this, the insurance company only pays that amount exceeding (the deductible amount
Consider a post office that is run by two clerks. Suppose that when Mr. Smith enters the system he discovers that Mr. Jones is being served by one of the clerks and Mr. Brown by the other. Suppose
Suppose that the amount of time one spends in a bank is exponentially distributed with mean ten minutes, that is, λ = 1 10 . What is the probability that a customer will spend more than fifteen
Applying the proposition to Example 2.23 yields E[X2] = 02(0.2) + (12)(0.5) + (22)(0.3) = 1.7 which, of course, checks with the result derived in Example 2.23.
6. Suppose five fair coins are tossed. Let E be the event that all coins land heads. Define the random variable IEFor what outcomes in the original sample space does IE equal 1? What is P{IE = 1}? IE
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.
8. Suppose the distribution function of X is given byWhat is the probability mass function of X? 0, b
9. If the distribution function of F is given bycalculate the probability mass function of X. F(b)= 9 10' 1, b
10. Suppose three fair dice are rolled. What is the probability at most one six appears?
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
12. Ona 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?
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
14. Suppose X has a binomial distribution with parameters 6 and 12. Show that X = 3 is the most likely outcome.
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
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
17. Suppose that an experiment can result in one of r possible outcomes, the ith outcome having probability pi , i = 1, . . . , r,ri=1 pi = 1. If n of these experiments are performed, and if the
18. Show that when r = 2 the multinomial reduces to the binomial.
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?
20. A television store owner figures that 50 percent of the customers entering his store will purchase an ordinary television set, 20 percent will purchase a color television set, and 30 percent will
21. In Exercise 20, what is the probability that our store owner sells three or more televisions on that day?
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
Let X be normally distributed with parameters μ and σ2. Find Var(X).
Calculate Var(X) when X represents the outcome when a fair die is rolled.
At a party N men throw their hats into the center of a room. The hats are mixed up and each man randomly selects one. Find the expected number of men who select their own hats.
Suppose there are 25 different types of coupons and suppose that each time one obtains a coupon, it is equally likely to be any one of the 25 types.Compute the expected number of different types that
The joint density function of X,Y is(a) Verify that the preceding is a joint density function.(b) Find Cov (X,Y). f(x, y) = e-(y+x/y), 0 < x, y < y
Compute the variance of a binomial random variable X with parameters n and p.
IfX and Y are independent gamma random variables with parameters(α, λ) and (β, λ), respectively, compute the joint density of U = X + Y and V = X/(X + Y).
If X and Y are independent binomial random variables with parameters (n, p) and (m, p), respectively, then what is the distribution of X + Y?
Calculate the distribution of X + Y when X and Y are independent Poisson random variables with means λ1 and λ2, respectively.
Show that if X and Y are independent normal random variables with parameters (μ1, σ2 1 ) and(μ2, σ2 2 ), respectively, then X + Y is normal with mean μ1 + μ2 and varianceσ2 1+ σ2 2 .
Let Xi, i = 1, 2, . . . , 10 be independent random variables, each being uniformly distributed over (0, 1). Estimate P{10 1 Xi > 7}.
The lifetime of a special type of battery is a random variable with mean 40 hours and standard deviation 20 hours. A battery is used until it fails, at which point it is replaced by a new one.
Consider a particle that moves along a set ofm + 1 nodes, labeled 0, 1, . . . ,m, that are arranged around a circle (see Figure 2.3). At each step the particle is equally likely to move one position
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
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?
22. If a fair coin is successively flipped, find the probability that a head first appears on the fifth trial.
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 probabilityThis is known as
24. The probability mass function of X is given byGive 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
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
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
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 ,ki=1 pi = 1. Each time a key is put in a nonempty box, we say that
46. If X is a nonnegative integer valued random variable, show thatHint: Define the sequence of random variables In, n ≥ 1, byNow express X in terms of the In.(b) If X and Y are both nonnegative
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
48. If X is a nonnegative random variable, and g is a differential function with g(0) = 0, thenProve the preceding when X is a continuous random variable. E[g(X)] = P(X > t)g'(t)dt
49. Prove that E[X2] ≥ (E[X])2. When do we have equality?
50. Let c be a constant. Show that(a) Var(cX) = c2Var(X);(b) Var(c + X) = Var(X).
51. A coin, having probability p of landing heads, is flipped until a head appears for the r th time. Let N denote the number of flips required. Calculate E[N].Hint: There is an easy way of doing
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.
53. If X is uniform over (0, 1), calculate E[Xn] and Var(Xn).
54. Let X and Y each take on either the value 1 or −1. Let p(1, 1) = P(X = 1, Y = 1}, p(1, 1) = P(X = 1, Y=-1), p(-1, 1) = P(X=-1, Y = 1}, p(-1,-1) = P(X = -1, Y=-1} Suppose that E[X]=E[Y]=0. Show
55. Suppose that the joint probability mass function of X and Y is(a) Find the probability mass function of Y.(b) Find the probability mass function of X.(c) Find the probability mass function of Y
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
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 + m,
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).
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.
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.
26. Find the expected number of games that are played when(a) i = 2;(b) i = 3.In both cases, show that this number is maximized when p = 1/2.
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
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
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
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
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 =
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? f(x) = c(1-x), 10, -1
34. Let the probability density of X be given by(a) What is the value of c? f(x) = Sc(4x-2x), 10, 0
35. The density of X is given byWhat is the distribution of X? Find P{X > 20}. f(x)= = {10/x, for x > 10 for x 10
36. A point is uniformly distributed within the disk of radius 1. That is, its density isFind the probability that its distance from the origin is less than x, 0 ≤ x ≤ 1. f(x, y) = C, 0 x + y 1
37. Let X1, X2, . . . ,Xn be independent random variables, each having a uniform distribution over (0,1). LetM = maximum (X1, X2, . . . ,Xn). Show that the distribution function of M, FM(·), is
38. If the density function of X equalsfindc. What is P{X > 2}? f(x) = Ice-2x 10, 0
39. The random variable X has the following probability mass function:Calculate E[X]. p(1) =, p(2) = 1, p(24) =
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
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 continuous
(a) If X is a discrete random variable with probability mass function p(x), then for any real-valued function g,(b) If X is a continuous random variable with probability density function f (x), then
Suppose cards numbered one through ten are placed in a hat, mixed up, and then one of the cards is drawn. If we are told that the number on the drawn card is at least five, then what is the
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 = EF ∪ EFc, E ∪ F = E ∪ FEc.
16. Use Exercise 15 to show that P(E ∪ F) = 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 colorblind 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(E1E2 · · ·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

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