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bayesian statistics an introduction

Probability & Statistics For Engineers And Scientists With R 1st Edition Michael Akritas - Solutions

6. The mean weight of frozen yogurt cups in an ice cream parlor is 8 oz. Suppose the weight of each cup served is normally distributed with standard deviation 0.5 oz, independently of others.(a) What is the probability of getting a cup weighing more than 8.64 oz?(b) What is the probability of
5. The yield strength (ksi) for A36 steel is normally distributed with μ = 43 and σ = 4.5.(a) What is the 25th percentile of the distribution of A36 steel strength?(b) What strength value separates the strongest 10% from the others?(c) What is the value of c such that the interval (43 − c, 43
4. Use the R command set.seed(111); hist(rexp(10000), breaks=35, freq=F) to generate a sample of size 10,000 from the exponential(1) distribution and to plot its histogram, and the additional R command curve(dexp, 0, 8, add=T) to superimpose the exponential(1) PDF on the graph. Does the histogram
3. Justify that the no-aging or memoryless property of the exponential random variable X, stated in (3.5-3), can be equivalently restated as P(X ≤ s+t|X ≥ s) = 1 − exp{−λt}.
2. The number of wrongly dialed phone calls you receive can be modeled as a Poisson process with the rate of one per month.(a) Find the probability that it will take between two and three weeks to get the first wrongly dialed phone call.(b) Suppose that you have not received a wrongly dialed phone
1. The lifespan of a car battery averages six years.Suppose the battery lifespan follows an exponential distribution.(a) Find the probability that a randomly selected car battery will last more than four years.(b) Find the variance and the 95th percentile of the battery lifespan.(c) Suppose a
User log-ons to a college’s computer network can be modeled as a Poisson process with a rate of 10 per minute. If the system’s administrator begins tracking the number of log-ons at 10:00 a.m., find the probability that the first log-on recorded occurs between 10 and 20 seconds after that.
Suppose the useful life time, in years, of a personal computer (PC) is exponentially distributed with parameter λ = 0.25. A student entering a four-year undergraduate program inherits a two-year-old PC from his sister who just graduated. Find the probability the useful life time of the PC the
23. Let X(t) be a Poisson process with rate α.(a) Use words to justify that the events[X(t) = 1] ∩ [X(1) = 1] and[X(t) = 1] ∩ [X(1) − X(t) = 0]are the same(b) Use Proposition 3.4-2 to find the probability of the event in (a) when α = 2 and t = 0.6.(c) It is given that X(1) = 1, that is,
22. Let X be the random variable that counts the number of events in each of the following cases.(a) The number of fish caught by an angler in an afternoon.(b) The number of disabled vehicles abandoned on I95 in a year.(c) The number of wrongly dialed telephone numbers in a given city in an
21. Suppose that a simple random sample of 200 is taken from the shipment of 10,000 electronic components of Exercise 15, which contais 300 defective components, and let Y denote the number of defective components in the sample.(a) The random variable Y has a hypergeometric(M1,M2, n) distribution,
20. An engineer at a construction firm has a subcontract for the electrical work in the construction of a new office building. From past experience with this electrical subcontractor, the engineer knows that each light switch that is installed will be faulty with probability p = 0.002 independent
19. A typesetting agency used by a scientific journal employs two typesetters. Let X1 and X2 denote the number of errors committed by typesetter 1 and 2, respectively, when asked to typeset an article. Suppose that X1 and X2 are Poisson random variables with expected values 2.6 and 3.8,
18. During a typical Pennsylvania winter, I80 averages 1.6 potholes per 10 miles. A certain county is responsible for repairing potholes in a 30-mile stretch of the interstate.Let X denote the number of potholes the county will have to repair at the end of next winter.(a) The distribution of the
17. Structural loads are forces applied to a structure or its components. Loads cause stresses that can lead to structural failure. It has been suggested that the occurrence of live (or probabilistic) structural loads over time in aging concrete structures can be modeled by a Poisson process with a
16. A particular website generates income when people visiting the site click on ads. The number of people visiting the website is modeled as a Poisson process with rateα = 30 per second. Of those visiting the site, 60% click on an ad. Let Y denote the number of those who will click on an ad over
15. In a shipment of 10,000 of a certain type of electronic component, 300 are defective. Suppose that 50 components are selected at random for inspection, and let X denote the number of defective components found.(a) The distribution of the random variable X is (choose one)(i) binomial (ii)
14. In a study of a lake’s fish population, scientists capture fish from the lake, then tag and release them.Suppose that over a period of five days, 200 fish of a certain type are tagged and released. As part of the same study, 20 such fish are captured three days later. Let X denote the number
13. A distributor receives a new shipment of 20 iPods.He draws a random sample of five iPods and thoroughly inspects the click wheel of each of them. Suppose that the shipment contains three iPods with a malfunctioning click wheel. Let X denote the number of iPods with a defective click wheel in
12. Suppose that six of the 15 school buses in a particular school district have developed a slight defect since their last inspection (the steering wheel shakes when braking).Five buses are to be selected for thorough inspection. Let X denote the number of buses among the five that are inspected
11. In the context of Exercise 10, suppose that after five interruptions the process undergoes a major evaluation.Suppose also that inspections happen once every week.Let Y denote the number of weeks between successive major evaluations.(a) The random variable Y is (choose one)(i) binomial (ii)
10. Average run length. To control the quality of a manufactured product, samples of the product are taken at specified inspection time periods and a quality characteristic is measured for each product. If the average measurement falls below a certain predetermined threshold, the process is
9. Two athletic teams, A and B, play a best-of-five series of games (i.e., the first team to win three games is the overall winner). Suppose team A is the better team and will win any game with probability 0.6, independently from other games.(a) Find the probability that the better team will be the
8. In the context of quality control, a company manufacturing bike helmets decides that helmets be inspected until the fifth helmet having a particular type of flaw is found. The total number X of helmets inspected will be used to decide whether or not the production process is under control.
7. In the grafting context of Exercise 1, suppose that grafts are done one at a time and the process continues until the first failed graft. Let X denote the number of grafts up to and including the first failed graft.(a) The random variable X is (choose one)(i) binomial (ii) hypergeometric (iii)
6. Suppose that in order for the defendant to be convicted in a military court the majority of the nine appointed judges must enter a guilty vote. Assume that a judge enters a guilty vote with probability 0.1 or 0.9 if the defendant is innocent or guilty, respectively, independently of other
5. The probability that a letter will be delivered within three working days is 0.9. You send out 10 letters on Tuesday to invite friends for dinner. Only those who receive the invitation by Friday (i.e., within 3 working days) will come. Let X denote the number of friends who come to dinner.(a)
4. A test consists of 10 true-false questions. Suppose a student answers the questions by flipping a coin. Let X denote the number of correctly answered questions.(a) Give the expected value and variance of X.(b) Find the probability the student will answer correctly exactly 5 of the questions.(c)
3. A company sells small, colored binder clips in packages of 20 and offers a money-back guarantee if two or more of the clips are defective. Suppose a clip is defective with probability 0.01, independently of other clips. Let X denote the number of defective clips in a package of 20.(a) The
2. Suppose that 30% of all drivers stop at an intersection having flashing red lights when no other cars are visible.Of 15 randomly selected drivers coming to an intersection under these conditions, let X denote the number of those who stop.(a) The random variable X is (choose one)(i) binomial (ii)
1. Grafting, the uniting of the stem of one plant with the stem or root of another, is widely used commercially to grow the stem of one variety that produces fine fruit on the root system of another variety with a hardy root system. For example, most sweet oranges grow on trees grafted to the root
Let X(t) be a Poisson process with rate α. It is given that X(1) = n. Show that the conditional distribution of X(0.4) is binomial(n, 0.4). In words, if we know that n events occurred in the interval (0, 1], then the number of events that occurred in the interval (0, 0.4] is a binomial(n, 0.4)
People enter a department store according to a Poisson process with rate α per hour.It is known that 30% of those entering the store will make a purchase of $50.00 or more. Find the probability mass function of the number of customers who will make purchases of $50.00 or more during the next hour.
Continuous inspection of electrolytic tin plate yields on average 0.2 imperfections per minute. Find each of the following:(a) The probability of one imperfection in three minutes.(b) The probability of at least two imperfections in five minutes.(c) The probability of at most one imperfection in
Suppose the monthly suicide rate in a certain county is 1 per 100,000 people. Give an approximation to the probability that in a city of 500,000 in this county there will be no more than six suicides in the next month.
Suppose that a person taking Vitamin C supplements contracts an average of three colds per year and that this average increases to five colds per year for persons not taking Vitamin C supplements. Suppose further that the number of colds a person contracts in a year is a Poisson random variable.(a)
Two athletic teams, A and B, play a best-of-three series of games (i.e., the first team to win two games is the overall winner). Suppose team A is the stronger team and will win any game with probability 0.6, independently from other games. Find the probability that the stronger team will be the
Three electrical engineers toss coins to see who pays for coffee. If all three match, they toss another round. Otherwise the “odd person” pays for coffee.(a) Find the probability that a round of tossing will result in a match (that is, either three heads or three tails).(b) Let X be the number
Items are being inspected as they come off the production line until the third defective item is found. Let X denote the number of non-defective items found. If an item is defective with probability p = 0.1 independently of other items, find the mean value and variance of X and P(X = 15).
Physical traits such as eye color are determined from a pair of genes, with one gene inherited from the mother and one from the father. Each gene can be either dominant(D) or recessive (R). People with gene pairs (DD), (DR), and (RD) are alike in that physical trait. Assume that a child is equally
The probability that an electronic product will last more than 5500 time units is 0.1.Let X take the value 1 if a randomly selected product lasts more than 5500 time units and the value 0 otherwise. Find the mean value and variance of X.
9. Plumbing suppliers typically ship packages of plumbing supplies containing many different combinations of items such as pipes, sealants, and drains. Almost invariably there are one or more parts in the shipment that are not correct: the part may be defective, missing, not the one that was
8. The length of time X, in hours, that a statistics reference book on a two-hour reserve at the engineering library is checked out by a randomly selected student has PDFFor books returned after two hours, students are charged a fine of $2.00 plus 6 cents times the number of minutes past the two
7. The CDF function of the checkout duration, X, in a certain supermarket, measured in minutes, is F(x) = 0 for x ≤ 0, F(x) = 1 for x > 2, and F(x) =x2 4for x between 0 and 2 .(a) Find the median and the interquartile range of the checkout duration.(b) Find E(X) and σX. You may use R commands
6. Consider the context of Example 3.3-9 where there is a cost associated with either early (i.e., before 15 days)or late (i.e., after 15 days) completion of the project. In an effort to reduce the cost, the company plans to start working on the project five days after the project is
5. The life time X, in months, of certain equipment is believed to have PDF f (x) = (1/100)xe−x/10, x >0 and f (x) = 0, x ≤ 0.Using R commands for the needed integrations, find E(X) and σ2X.
4. A metal fabricating plant currently has five major pieces under contract, each with a deadline for completion.Let X be the number of pieces completed by their deadlines. Suppose that X is a random variable with PMF p(x) given by(a) Compute the expected value and variance of X.(b) For each piece
3. A customer entering an electronics store will buy a flat screen TV with probability 0.3. Sixty percent of the customers buying a flat screen TV will spend $750.00 and 40% will spend $400.00. Let X denote the amount spent on flat screen TVs by two random customers entering the store.(a) Find the
2. Let X have PMF(a) Calculate E(X) and E(1/X). (b) In a win-win game, the player will win a monetary prize, but has to decide between the fixed price of $1000/E(X) and the random price of $1000/X, where the random variable X has the PMF given above.Which choice would you recommend the player make?
1. A simple random sample of three items is selected from a shipment of 20 items of which four are defective.Let X be the number of defective items in the sample.(a) Find the PMF of X.(b) Find the mean value and variance of X.
Let X have PDF f (x) = 0.001e−0.001x for x ≥ 0, and f (x) = 0 for x < 0. Find a general expression for the 100(1−α)-th percentile of X in terms of α, for α between 0 and 1, and use it to find the interquartile range of X.
Suppose X has PDF f (x) = e−x for x ≥ 0, and f (x) = 0 for x < 0. Find the median and the 95th percentile of X.
Let Y ∼ U(A,B), that is, Y has the uniform in [A,B] distribution (see Example 3.2-5). Show that Var(Y) = (B − A)2/12.
Let X have PDF fX(x) = 0.1 exp(−0.1x) for x > 0 and 0 otherwise. Find the variance and standard deviation of X.
Let X ∼ U(0, 1), that is, X has the uniform in [0, 1] distribution (see Example 3.2-4).Show Var(X) = 1/12.
Consider the experiment where product items are being inspected for the presence of a particular defect until the first defective product item is found. Let X denote the total number of items inspected. Suppose a product item is defective with probability p, p>0, independently of other product
Roll a die and let X denote the outcome. Find Var(X).
Select a product from the production line and let X take the value 1 or 0 as the product is defective or not. If p is the probability that the selected item is defective, find Var(X) in terms of p.
The time T, in days, required for the completion of a contracted project is a random variable with PDF fT(t) = 0.1 exp(−0.1t) for t > 0 and 0 otherwise. Suppose the contracted project must be completed in 15 days. If T 15 there is a cost of $10(T − 15). Find the expected value of the cost.
Let Y ∼ U(A,B), that is, Y has the uniform in [A,B] distribution (see Example 3.2-5). Show that E(Y) = (B + A)/2.
A book store purchases three copies of a book at $6.00 each and sells them for $12.00 each. Unsold copies are returned for $2.00 each. Let X = {number of copies sold}and Y = {net revenue}. If the PMF of X isfind the expected value of Y. X 0 1 2 3 Px(x) 0.1 0.2 0.2 0.5
The time T, in days, required for the completion of a contracted project is a random variable with PDF fT(t) = 0.1 exp(−0.1t) for t > 0 and 0 otherwise. Find the expected value of T.
Let X ∼ U(0, 1), that is, X has the uniform in [0, 1] distribution (see Example 3.2-4).Show that E(X) = 0.5.
If the PDF of X is f (x) = 2x for 0 ≤ x ≤ 1 and 0 otherwise, find E(X).
Consider the experiment where product items are being inspected for the presence of a particular defect until the first defective product item is found. Let X denote the total number of items inspected. Suppose a product item is defective with probability p, p>0, independently of other product
Select a product item from the production line and let X take the value 1 or 0 as the product item is defective or not. Let p be the proportion of defective items in the conceptual population of this experiment. Find E(X) in terms of p.
Suppose the population of interest is a batch of N = 100 units, 10 of which have some type of defect, received by a distributor. An item is selected at random from the batch and is inspected. Let X take the value 1 if the selected unit has the defect and 0 otherwise. Use formula (3.3.1) to compute
11. Use the R commands set.seed(111); hist(runif(100), freq=F) to generate a sample of size 100 from the uniform(0, 1) distribution and to plot its histogram, and the additional R command curve(dunif, 0, 1, add=T) to superimpose the uniform(0, 1) PDF on the graph. Does the histogram provide a
10. The time X in hours for a certain plumbing manufacturer to deliver a custom made fixture is a random variable with PDFAn architect overseeing a renovation orders a custom made fixture to replace the old one, which unexpectedly broke. If the ordered fixture arrives within three days no
9. In a game of darts, a player throws the dart and wins X = 30/D dollars, where D is the distance in inches of the dart from the center of the dartboard. Suppose a player throws the dart in such a way that it lands in a randomly selected point on the 18-inch diameter dartboard. Thus, the
8. The cumulative distribution function of checkout duration X, measured in minutes, in a certain supermarket is F(x) =x2 4for x between 0 and 2, F(x) = 0 for x ≤ 0, and F(x) = 1 for x > 2.(a) Find the probability that the duration is between 0.5 and 1 minute.(b) Find the probability density
7. Let X ∼ U(0, 1), and set Y = −log(X). Give the sample space of Y, and find the CDF and PDF of Y. (Hint.FY(y) = P(Y ≤ y) = P(X ≥ exp(−y)).)
6. Let X ∼ U(0, 1). Show that Y = 3+6X ∼ U(3, 9), that is, that it has the uniform in [3, 9] distribution defined in Example 3.2-5. (Hint. Find the CDF of Y and show it has the form of the CDF found in the solution of Example 3.2-5.)
5. Answer the following questions.(a) Check whether or not each of f1(x), f2(x) is a legitimate probability density function(b) Let X denote the resistance of a randomly chosen resistor, and suppose that its PDF is given by(i) Find k and the CDF of X, and use the CDF to calculate P(8.6 ≤ X ≤
4. A simple random sample of size n = 3 is drawn from a batch of ten product items. If three of the 10 items are defective, find the PMF and the CDF of the random variable X = {number of defective items in the sample}.
3. Let Y denote the cost, in hundreds of dollars, incurred to the metal fabricating plant of Exercise 2 due to missing deadlines. Suppose the CDF of Y is(a) Plot the CDF and find the probability that the cost from delays will be at least $200.00.(b) Find the probability mass function of Y. 0 y
2. A metal fabricating plant currently has five major pieces under contract each with a deadline for completion.Let X be the number of pieces completed by their deadlines, and suppose its PMF p(x) is given by(a) Find and plot the CDF of X.(b) Use the CDF to find the probability that between one and
1. Answer the following questions.(a) Check whether or not each of p1(x), p2(x) is a legitimate probability mass function.(b) Find the value of the multiplicative constant k so p(x)given in the following table is a legitimate probability mass function. X 0 1 2 3 P1(x) 0.3 0.3 0.5 -0.1 * x P2(x) 0.1
Suppose that a point is selected at random from a circle centered at the origin and having radius 6. Thus, the probability of the point lying in a region A of this circle is proportional to the area of A. Find the PDF of the distance D of this point from the origin.
Let X denote the amount of time a statistics reference book on a two-hour reserve at the engineering library is checked out by a randomly selected student. Suppose that X has density functionFor books returned after two hours, students are charged a fine of $2.00 plus $1.00 for each additional
In the context of Example 3.2-6, let(T be the life time, measured in minutes, of the randomly selected electrical component. Find the PDF of(T.
If the life time T, measured in hours, of a randomly selected electrical component has PDF fT(t) = 0 for t < 0, and fT(t) = 0.001 exp(−0.001t) for t ≥ 0, find the probability the component will last between 900 and 1200 hours of operation.
A random variable X is said to have the uniform in [A,B] distribution, denoted by X ∼ U(A,B), if its PDF isFind the CDF F(x). 0 if x < A 1 f(x) = if A x B B-A 0 if x > B.
If X ∼ U(0, 1), show that the PDF of X is 0 if x < 0 fx(x)=1 if 0x1 0 if x 1. >
The PMF of a random variable X isFind the CDF F of X. x 1 2 3 4 p(x) 0.4 0.3 0.2 0.1
11. Find the reliability of a 3-out-of-4 system if each of its four components functions with probability 0.9 independently of the others.
10. The system of components shown in Figure 2-15 below functions as long as components 1 and 2 both function or components 3 and 4 both function. Each of the four components functions with probability 0.9 independently of the others. Find the probability that the system functions. 1 2 3 4 Figure
9. Show that if E1,E2,E3 are independent, then E1 is independent from E2∪E3. (Hint. By the Distributive Law, P(E1 ∩ (E2 ∪ E3)) = P((E1 ∩ E2) ∪ (E1 ∩ E3)). Using the formula for the probability of the union of two events(part (1) of Proposition 2.4-2) and the independence of E1,E2,E3,
8. Roll a die twice and record the two outcomes. Let E1 = {the sum of the two outcomes is 7}, E2={the outcome of the first roll is 3}, E3={the outcome of the second roll is 4}. Show that E1, E2, E3 are pairwise independent but (2.6.4) does not hold.
7. Some information regarding the composition of the student athlete population in the high school mentioned in Exercise 6 is given in the table below. For example, 65% of the student athletes are male, 50% of the student athletes play basketball, and female athletes do not play football. For a
6. An athlete is selected at random from the population of student athletes in a small private high school, and the athlete’s gender and sports preference is recorded.Define the events M = {the student athlete is male}, F= {the student athlete is female}, and T= {the student athlete prefers
5. Quality control engineers monitor the number of nonconformances per car in an automobile production facility.Each day, a simple random sample of four cars from the first assembly line and a simple random sample of three cars from the second assembly line are inspected.The probability that an
4. An experiment consists of inspecting fuses as they come off a production line until the first defective fuse is found. Assume that each fuse is defective with a probability of 0.01, independently of other fuses.Find the probability that a total of eight fuses are inspected.
3. A simple random sample of 10 software widgets are chosen for installation. If 10% of this type of software widgets have connectivity problems, find the probability of each of the following events.(a) None of the 10 have connectivity problems.(b) The first widget installed has connectivity
2. In the context of Exercise 2.5-3, are the events [X = 1]and [Y = 1] independent? Justify your answer.
1. In a batch of 10 laser diodes, two have efficiency below 0.28, six have efficiency between 0.28 and 0.35, and two have efficiency above 0.35. Two diodes are selected at random and without replacement. Are the events E1 = {the first diode selected has efficiency below 0.28} and E2 = {the second
Find the reliability of a 2-out-of-3 system whose three components function with probabilities p1 = 0.9, p2 = 0.85, and p3 = 0.8, respectively, independently of each other.
The three components of the parallel system shown in the right panel of Figure 2-14 function with probabilities p1 = 0.9, p2 = 0.85, and p3 = 0.8, respectively, independently of each other. What is the probability the system functions?
The three components of the series system shown in the left panel of Figure 2-14 fail with probabilities p1 = 0.1, p2 = 0.15, and p3 = 0.2, respectively, independently of each other. What is the probability the system will fail?
At 25oC, 20% of a certain type of laser diodes have efficiency below 0.3 mW/mA.For five diodes, selected by simple random sampling from a large population of such diodes, find the probability of the following events.(a) All five have efficiency above 0.3 at 25oC.(b) Only the second diode selected

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