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statistics for engineers and scientists

Introduction To Probability And Statistics For Engineers And Scientists 6th Edition Sheldon M Ross - Solutions

44. If X and Y are independent chi-square random variables with 3 and 6 degrees of freedom, respectively, determine the probability that X + Y will exceed 10.
43. If X is a chi-square random variable with 6 degrees of freedom, finda. P{X ≤ 6};b. P{3 ≤ X ≤ 9}.
42.11 When shooting at a target in a two-dimensional plane, suppose that the horizontal miss distance is normally distributed with mean 0 and variance 4 and is independent of the vertical miss distance, which is also normally distributed with mean 0 and variance 4. Let D denote the distance between
41.10 Earthquakes occur in a given region in accordance with a Poisson process with rate 5 per year.a. What is the probability there will be at least two earthquakes in the first half of 2015?b. Assuming that the event in part (a) occurs, what is the probability that there will be no earthquakes
40.9 Let X1,X2, . . . , Xn denote the first n interarrival times of a Poisson process and set Sn =n i=1Xi .a. What is the interpretation of Sn?b. Argue that the two events {Sn ≤ t} and {N(t) ≥ n} are identical.c. Use part (b) to show thatd. By differentiating the distribution function of Sn
39. Jones figures that the total number of thousands of miles that a used auto can be driven before it would need to be junked is an exponential random variable with parameter 1 20 . Smith has a used car that he claims has been driven only 10,000 miles. If Jones purchases the car, what is the
38. The number of years a radio functions is exponentially distributed with parameter λ = 1 8 . If Jones buys a used radio, what is the probability that it will be working after an additional 10 years?
37. The time (in hours) required to repair a machine is an exponentially distributed random variable with parameter λ = 1.a. What is the probability that a repair time exceeds 2 hours?b. What is the conditional probability that a repair takes at least 3 hours, given that its duration exceeds 2
36. An IQ test produces scores that are normally distributed with mean value 100 and standard deviation 14.2. The top 1 percent of all scores are in what range?
35. The height of adult women in the United States is normally distributed with mean 64.5 inches and standard deviation 2.4 inches. Find the probability that a randomly chosen woman isa. less than 63 inches tall;b. less than 70 inches tall;c. between 63 and 70 inches tall.d. Alice is 72 inches
34. The annual rainfall in Cincinnati is normally distributed with mean 40.14 inches and standard deviation 8.7 inches.a. What is the probability this year’s rainfall will exceed 42 inches?b. What is the probability that the sum of the next 2 years’ rainfall will exceed 84 inches?c. What is the
33. Value at risk (VAR) has become a key concept in financial calculations.The VAR of an investment is defined as that value v such that there is only a 1 percent chance that the loss from the investment will exceed v.a. If the gain from an investment is a normal random variable with mean 10 and
32. The sample mean and sample standard deviation on your economics examination were 60 and 20, respectively; the sample mean and sample standard deviation on your statistics examination were 55 and 10, respectively.You scored 70 on the economics exam and 62 on the statistics exam. Assuming that
31. The salaries of pediatric physicians are approximately normally distributed.If 25 percent of these physicians earn below 180, 000 and 25 percent earn above 320, 000, what fraction earna. below 250, 000;b. between 260, 000 and 300, 000?
30. A random variable X is said to have a lognormal distribution if log X is normally distributed. If X is lognormal with E[logX] = μ and Var(logX) = σ2, determine the distribution function of X. That is, what is P{X ≤ x}?
29. Leta. Show that for any μ and σ 1=fe-x/2dx.
28. A manufacturer produces bolts that are specified to be between 1.19 and 1.21 inches in diameter. If its production process results in a bolt’s diameter being normally distributed with mean 1.20 inches and standard deviation .005, what percentage of bolts will not meet specifications?
27. Let X be normal with mean μ and variance σ2. For fixed μ, show that P(X >10) is an increasing function of σ when μ10. Give an intuitive reason why the preceding is true.
26. The weekly demand for a product approximately has a normal distribution with mean 1000 and standard deviation 200. The current on hand inventory is 2200 and no deliveries will be occurring in the next two weeks. Assuming that the demands in different weeks are independent,a. what is the
25. The annual rainfall (in inches) in a certain region is normally distributed with μ = 40, σ = 4. What is the probability that in 2 of the next 4 years the rainfall will exceed 50 inches? Assume that the rainfalls in different years are independent.
24. The Scholastic Aptitude Test mathematics test scores across the population of high school seniors follow a normal distribution with mean 500 and standard deviation 100. If five seniors are randomly chosen, find the probability that (a) all scored below 600 and (b) exactly three of them scored
23. Let X1 and X2 be independent normal random variables, each having mean 10 and variance σ2. Which probability is larger:a. P(X1 > 15) or P(X1 +X2 > 25);b. P(X1 > 15) or P(X1 +X2 > 30)?c. Find x such that P(X1 + X2 >x) = P(X1 > 15).
22. You arrive at a bus stop at 10 o’clock, knowing that the bus will arrive at some time uniformly distributed between 10 and 10:30. What is the probability that you will have to wait longer than 10 minutes? If at 10:15 the bus has not yet arrived, what is the probability that you will have to
21. If U is uniformly distributed on (0, 1), show that a+(b−a)U is uniform on (a, b).
20. Independent trials, each of which is a success with probability p, are successively performed. Let X denote the first trial resulting in a success. That is, X will equal k if the first k−1 trials are all failures and the kth a success.X is called a geometric random variable. Computea. P{X =
19. Let X denote a hypergeometric random variable with parameters n, m, and k. That is, (0)(2) m P{X=i}=- n+m i=0,1,..., min(k,n) a. Derive a formula for P{X=i} in terms of P{X = i-1}. b. Use part (a) to compute P{X=i) for i = 0, 1, 2, 3, 4, 5 when n=m= 10, k = 5, by starting with P{X=0}. c. Based
18. A contractor purchases a shipment of 100 transistors. It is his policy to test 10 of these transistors and to keep the shipment only if at least 9 of the 10 are in working condition. If the shipment contains 20 defective transistors, what is the probability it will be kept?
17. If X is a Poisson randomvariable with mean λ, show that P{X = i } first increases and then decreases as i increases, reaching its maximum value when i is the largest integer less than or equal to λ.
16. The probability of error in the transmission of a binary digit over a communication channel is 1/103. Write an expression for the exact probability of more than 3 errors when transmitting a block of 103 bits. What is its approximate value? Assume independence.
15. The game of frustration solitaire is played by turning the cards of a randomly shuffled deck of 52 playing cards over one at a time. Before you turn over the first card, say ace; before you turn over the second card, say two; before you turn over the third card, say three. Continue in this
14. Approximately 80,000 marriages took place in the state of New York last year. Estimate the probability that for at least one of these couplesa. both partners were born on April 30;b. both partners celebrated their birthday on the same day of the year.State your assumptions.
13. In the 1980s, an average of 121.95 workers died on the job each week.Give estimates of the following quantities:a. the proportion of weeks having 130 deaths or more;b. the proportion of weeks having 100 deaths or less.Explain your reasoning.
12. The number of times that an individual contracts a cold in a given year is a Poisson random variable with parameter λ = 3. Suppose a new wonder drug (based on large quantities of vitamin C) has just been marketed that reduces the Poisson parameter to λ = 2 for 75 percent of the population.For
11. 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) exactly once, and (c) at least twice?
10. Compare the Poisson approximation with the correct binomial probability for the following cases:a. P{X = 2} when n = 10, p = .1;b. P{X = 0} when n = 10, p = .1;c. P{X = 4} when n = 9, p = .2.
9. An experiment has possible outcomes 1, 2, . . . , r, with i being the outcome with probability pi , r i=1 pi = 1. Suppose there are n independent replications of this experiment, and let Ni be the number of them that result in outcome i, i = 1, . . . , r.a. What is the distribution of N1?b. Are
8. If X is a binomial random variable with parameters n and p, where 0 p b. As k goes from 0 to n, P{X = k} first increases and then decreases, reaching its largest value when k is the largest integer less than or equal to (n+ 1)p. Pn-k a. P{X=k+1}= 1-pk+1 P{X=k}, k=0,1,...,n-1.
7. If X and Y are binomial random variables with respective parameters(n,p) and (n, 1−p), verify and explain the following identities:a. P{X ≤ i} = P{Y ≥ n− i};b. P{X = k} = P{Y = n− k}.
6. Let X be a binomial random variable with E[X] = 7 and Var(X) = 2.1 Finda. P{X = 4};b. P{X>12}.
5. At least one-half of an airplane’s engines are required to function in order for it to operate. If each engine independently functions with probability p, for what values of p is a 4-engine plane more likely to operate than a 2-engine plane?
4. Suppose that a particular trait (such as eye color or left-handedness) of a person is classified on the basis of one pair of genes, and suppose that d represents a dominant gene and r a recessive gene. Thus, a person with dd genes is pure dominance, one with rr is pure recessive, and one with rd
3. If each voter is for Proposition A with probability .7, what is the probability that exactly 7 of 10 voters are for this proposition?
2. A communications channel transmits the digits 0 and 1. However, due to static, the digit transmitted is incorrectly received with probability .2.Suppose that we want to transmit an important message consisting of one binary digit. To reduce the chance of error, we transmit 00000 instead of 0 and
1. A satellite system consists of 4 components and can function adequately if at least 2 of the 4 components are in working condition. If each component is, independently, in working condition with probability .6, what is the probability that the system functions adequately?
57. Let X and Y have respective distribution functions FX and FY , and suppose that for some constants a and b >0, Fx(x) = Fr (5) () Fy a. Determine E[X] in terms of E[Y]. b. Determine Var(X) in terms of Var(Y). Hint: X has the same distribution as what other random variable?
56. From past experience, a professor knows that the test score of a student taking her final examination is a random variable with mean 75.a. Give an upper bound to the probability that a student’s test score will exceed 85.Suppose in addition the professor knows that the variance of a
55. Suppose that X is a random variable with mean and variance both equal to 20. What can be said about P{0 ≤ X ≤ 40}?
53. Suppose that X has density function f (x) = e−x, x>0 Compute the moment generating function of X and use your result to determine its mean and variance. Check your answer for the mean by a direct calculation.54. If the density function of X is f (x) = 1, 0
52. If X1 and X2 have the same probability distribution function, show that Cov(X1 − X2, X1 +X2) = 0 Note that independence is not being assumed.
51. In Example 4.5.f, compute Cov(Xi,Xj ) and use this result to show that Var(X) = 1.
50. Consider n independent trials, each of which results in any of the outcomes i, i = 1, 2, 3, with respective probabilities p1,p2,p3,3 i=1 pi = 1.Let Ni denote the number of trials that result in outcome i, and show that Cov(N1,N2)=−np1p2. Also explain why it is intuitive that this covariance
49. Let X have variance σ2 x and let Y have variance σ2 y . Starting with 0Var(X/ox+Y/y) show that -1 Corr(X, Y) Now using that 0Var(X/ox-Y/,) conclude that -1 Corr(X, Y) 1 Using the result that Var(Z) = 0 implies that Z is constant, argue that, if Corr(X, Y) = 1 or -1, then X and Y are related
48. Prove Equation (4.7.5) by using mathematical induction.
47. Verify Equation (4.7.4).
46. Find Corr(X1,X2) for the random variables of Problem 44.
45. A product is classified according to the number of defects it contains and the factory that produces it. Let X1 and X2 be the random variables that represent the number of defects per unit (taking on possible values of 0, 1, 2, or 3) and the factory number (taking on possible values 1 or 2),
44. Let Xi denote the percentage of votes cast in a given election that are for candidate i, and suppose that X1 and X2 have a joint density functiona. Find the marginal densities of X1 and X2.b. Find E[Xi ] and Var(Xi ) for i = 1, 2. fx1,x2(x, y) = { 3(x+y), if x0, y0,0x + y 1 0, if otherwise
43. A random variable X, which represents the weight (in ounces) of an article, has density function,a. Calculate the mean and variance of the random variable X.b. The manufacturer sells the article for a fixed price of $2.00. He guarantees to refund the purchase money to any customer who finds the
42. This problem refers to Example 4.6.c concerning the friendship paradox.Let W be a randomly chosen friend of the randomly chosen individual X. That is, W is equally likely to be any of the friends of X. It can be shown that E[f (W)] ≥ E[f (X)]. That is, the average number of friends of a
41. Compute the mean and variance of the number of heads that appear in 3 flips of a fair coin.
40. Let pi = P{X = i} and suppose that p1 + p2 + p3 = 1. If E[X] = 2, what values of p1,p2,p3 (a) maximize and (b) minimize Var(X)?
39. Suppose that X is equally likely to take on any of the values 1, 2, 3, 4.Compute (a) E[X] and (b) Var(X).
38. Compute the expectation and variance of the number of successes in n independent trials, each of which results in a success with probability p.Is independence necessary?
37. A community consists of 100married couples. If 50members of the community die, what is the expected number ofmarriages that remain intact?Assume that the set of people who die is equally likely to be any of the 200 50 groups of size 50. Hint: For i = 1,..., 100 let Xi = { 1 if neither member
36. We say that mp is the 100p percentile of the distribution function F if F(mp) = p Find mp for the distribution having density function f (x) = 2e−2x, x≥ 0
35. The median, like the mean, is important in predicting the value of a random variable. Whereas it was shown in the text that the mean of a random variable is the best predictor from the point of view of minimizing the expected value of the square of the error, the median is the best predictor if
34. If X is a continuous random variable having distribution function F, then its median is defined as that value of m for which F(m) = 1/2 Find the median of the random variables with density functiona. f (x) = e −x, x≥ 0;b. f (x) = 1, 0 ≤ x ≤ 1.
33. Ten balls are randomly chosen from an urn containing 17 white and 23 black balls. Let X denote the number of white balls chosen. Compute E[X]a. by defining appropriate indicator variables Xi, i = 1, .. ., 10 so thatb. by defining appropriate indicator variables Yi = 1, . . ., 17 so that 10 x=
32. If E[X] = 2 and E[X2] = 8, calculate (a) E[(2 + 4X)2] and (b) E[X2 +(X +1)2].
31. The time it takes to repair a personal computer is a random variable whose density, in hours, is given byThe cost of the repair depends on the time it takes and is equal to 40 +30 √x when the time is x. Compute the expected cost to repair a personal computer. f(x)= 0
30. Suppose that X has density functionCompute E[Xn] (a) by computing the density of Xn and then using the definition of expectation and (b) by using Proposition 4.5.1. f(x)= { 1 0 < x < 1 0 otherwise
29. Let X1, X2, . . . , Xn be independent random variables having the common density functionFind (a) E[Max(X1, . . . , Xn)] and (b) E[Min(X1, . . . , Xn)]. f(x)= 1 0 < x
28. The lifetime in hours of electronic tubes is a random variable having a probability density function given by f (x) = a2 x e−ax, x≥ 0 Compute the expected lifetime of such a tube.
27. The density function of X is given by a+bx2 0x1 f(x)= { 0 If E[X]=, find a, b. otherwise
26. Suppose that two teams play a series of games that end when one of them has won i games. Suppose that each game played is, independently, won by team A with probability p. Find the expected number of games that are played when i = 2. Also show that this number is maximized when p = 1 2 .
25. A total of 4 buses carrying 148 students from the same school arrive at a football stadium. The buses carry, respectively, 40, 33, 25, and 50 students.One of the students is randomly selected. Let X denote the number of students that were on the bus carrying this randomly selected student.One
24. An insurance company writes a policy to the effect that an amount of money A must be paid if some event E occurs within a year. If the company estimates that E will occur within a year with probability p, what should it charge the customer so that its expected profit will be 10 percent of A?
23. Each night different meteorologists give us the “probability” that it will rain the next day. To judge how well these people predict, we will score each of them as follows: If a meteorologist says that it will rain with probability p, then he or she will receive a score of 1−(1 −p)2 if
22. Compute the expected value of the random variable in Problem 3.
21. Compute the expected value of the random variable in Problem 1.
20. Show that X and Y are independent if and only ifa. pX|Y(x|y) = pX(x) in the discrete caseb. fX|Y(x|y) = fX(x) in the continuous case
19. Compute the conditional density function of X given Y = y in (a) Problem 10 and (b) Problem 13.
18. In Example 4.3.b, determine the conditional probability mass function of the size of a randomly chosen family containing 2 girls.
17. When a current I (measured in amperes) flows through a resistance R(measured in ohms), the power generated (measured in watts) is given by W = I 2R. Suppose that I and R are independent random variables with densities fI (x) = 6x(1− x) 0 ≤ x ≤ 1 fR(x) = 2x 0 ≤ x ≤ 1 Determine the
16. Suppose that X and Y are independent continuous random variables.Show that a. P{X+Ya) = Fx (a y) fr (y) dy -00 b. P(X Y) = Fx(y) fr(y) dy -00 where fy is the density function of Y, and Fx is the distribution function of X.
15. Is Problem 14 consistent with the results of Problems 12 and 13?
14. If the joint density function of X and Y factors into one part depending only on x and one depending only on y, show that X and Y are independent.That is, if f (x, y) = k(x)h( y), −∞
13. The joint density of X and Y isa. Compute the density of X.b. Compute the density of Y.c. Are X and Y independent? f(x, y) = 2 0
12. The joint density of X and Y is given bya. Compute the density of X.b. Compute the density of Y.c. Are X and Y independent? xe-(x+y) x>0,y>0 f(x, y) = 0 otherwise
11. 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 is given by FM(x) = xn, 0 ≤ x ≤ 1 What is the probability density function of M?
10. The joint probability density function of X and Y is given by f(x, y) = (x+), 0
9. A set of five transistors are to be tested, one at a time in a randomorder, to see which of them are defective. Suppose that three of the five transistors are defective, and let N1 denote the number of tests made until the first defective is spotted, and let N2 denote the number of additional
8. If the density function of X equalsfindc. What is P{X>2}? ce-2x 0
7. The lifetime in hours of a certain kind of radio tube is a random variable having a probability density function given byWhat is the probability that exactly 2 of 5 such tubes in a radio set will have to be replaced within the first 150 hours of operation? Assume that the events Ei, i = 1, 2, 3,
6. The amount of time, in hours, that a computer functions before breaking down is a continuous random variable with probability density function given byWhat is the probability that a computer will function between 50 and 150 hours before breaking down? What is the probability that it will
5. Suppose the random variable X has probability density functiona. Find the value of c.b. Find P{.4 cx, if 0x1 f(x)= 0, otherwise
4. The distribution function of the random variable X is given F(x)= x 0223 x
3. In Problem 2, if the coin is assumed fair, for n = 3, 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. Five men and 5 women are ranked according to their scores on an examination.Assume that no two scores are alike and all 10! possible rankings are equally likely. Let X denote the highest ranking achieved by a woman(for instance, X = 2 if the top-ranked person was male and the nextranked person

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