New Semester
Started
Get
50% OFF
Study Help!
--h --m --s
Claim Now
Question Answers
Textbooks
Find textbooks, questions and answers
Oops, something went wrong!
Change your search query and then try again
S
Books
FREE
Study Help
Expert Questions
Accounting
General Management
Mathematics
Finance
Organizational Behaviour
Law
Physics
Operating System
Management Leadership
Sociology
Programming
Marketing
Database
Computer Network
Economics
Textbooks Solutions
Accounting
Managerial Accounting
Management Leadership
Cost Accounting
Statistics
Business Law
Corporate Finance
Finance
Economics
Auditing
Tutors
Online Tutors
Find a Tutor
Hire a Tutor
Become a Tutor
AI Tutor
AI Study Planner
NEW
Sell Books
Search
Search
Sign In
Register
study help
business
introduction to probability statistics
Probability And Statistics For Engineering And The Sciences 8th Edition Jay L Devore, Roger Ellsbury - Solutions
9. Component i is said to be relevant to the system if for some state vector x, Otherwise, it is said to be irrelevant.(a) Explain in words what it means for a component to be irrelevant.(b) Let be the minimal path sets of a system, and let S denote the set of components. Show that if and only if
10. Let denote the time of failure of the ith component; let denote the time to failure of the system ϕ as a function of the vector . Show that where are the minimal cut sets, and the minimal path sets.
11. Give the reliability function of the structure of Exercise 8.
*12. Give the minimal path sets and the reliability function for the structure in Fig. 9.22.
13. Let be the reliability function. Show that
14. Compute the reliability function of the bridge system (see Fig.9.11) by conditioning upon whether or not component 3 is working.
15. Compute upper and lower bounds of the reliability function (using Method 2) for the systems given in Exercise 4, and compare them with the exact values when .
16. Compute the upper and lower bounds of using both methods for the(a) two-out-of-three system and(b) two-out-of-four system.(c) Compare these bounds with the exact reliability when(i)(ii)(iii)
*17. Let N be a nonnegative, integer-valued random variable. Show that and explain how this inequality can be used to derive additional bounds on a reliability function.Now multiply both sides by .
18. Consider a structure in which the minimal path sets are {1, 2, 3}and {3, 4, 5}.(a) What are the minimal cut sets?(b) If the component lifetimes are independent uniform () random variables, determine the probability that the system life will be less than .
19. Let denote independent and identically distributed random variables and define the order statistics by Show that if the distribution of is IFR, then so is the distribution of
20. Let F be a continuous distribution function. For some positive α,define the distribution function G by Find the relationship between and , the respective failure rate functions of G and F.
34. For the tandem queue model verify that satisfies the balance Eqs. (8.15).
*32. Let D denote the time between successive departures in a stationary queue withI mage. Show, by conditioning on whether or not a departure has left the system empty, that D is exponential with rate λ.Hint: By conditioning on whether or not the departure has left the system empty we see that
34. An queueing system is cleaned at the fixed timesI mage All customers in service when a cleaning begins are forced to leave early and a cost is incurred for each customer. Suppose that a cleaning takes timeI mage, and that all customers who arrive while the system is being cleaned are lost, and
50. Consider a semi-Markov process in which the amount of time that the process spends in each state before making a transition into a different state is exponentially distributed. What kind of process is this?
51. In a semi-Markov process, letI mage denote the conditional expected time that the process spends in state i given that the next state is j.(a) Present an equation relating to the Image.(b) Show that the proportion of time the process is in i and will next enter j is equal to Image.Hint: Say
52. A taxi alternates between three different locations. Whenever it reaches location i, it stops and spends a random time having mean Image before obtaining another passenger, . A passenger entering the cab at location i will want to go to location j with probability . The time to travel from i to
*53. Consider a renewal process having the gamma interarrival distribution, and let denote the time from t until the next renewal. Use the theory of semi-Markov processes to show that whereI mage is the gammaI mage distribution function.
54. To prove Eq. (7.24), define the following notation:Image In terms of this notation, write expressions for(a) the amount of time during the first m transitions that the process is in state i;(b) the proportion of time during the first m transitions that the process is in state i.Argue that, with
55. In 1984 the country of Morocco in an attempt to determine the average amount of time that tourists spend in that country on a visit tried two different sampling procedures. In one, they questioned randomly chosen tourists as they were leaving the country; in the other, they questioned randomly
56. In Example 7.20, show that if F is exponential with rate μ, then That is, when buses arrive according to a Poisson process, the average number of people waiting at the stop, averaged over all time, is equal to the average number of passengers waiting when a bus arrives. This may seem
57. If a coin that comes up heads with probability p is continually flipped until the pattern HTHTHTH appears, find the expected number of flips that land heads.
58. Let , be independent random variables withI mage. IfI mage,Image, find the expected time and the variance of the number of variables that need be observed until the pattern 1, 2, 3, 1, 2 occurs.
59. A coin that comes up heads with probability 0.6 is continually flipped. Find the expected number of flips until either the sequence Image or the sequence ttt occurs, and find the probability that ttt occurs first.
60. Random digits, each of which is equally likely to be any of the digits 0 through 9, are observed in sequence.(a) Find the expected time until a run of 10 distinct values occurs.(b) Find the expected time until a run of 5 distinct values occurs.
61. LetI mage where are independent random variables having distribution function and T is independent of the and has probability mass functionI mage. Show thatI mage satisfies Eq.
1. For the queue, compute(a) the expected number of arrivals during a service period and(b) the probability that no customers arrive during a service period.Hint: “Condition.”
49. Consider a system that can be in either state 1 or 2 or 3. Each time the system enters state i it remains there for a random amount of time having mean and then makes a transition into state j with probability . Suppose(a) What proportion of transitions takes the system into state 1?(b) If
*48. In Example 7.20, let π denote the proportion of passengers that wait less than x for a bus to arrive. That is, with equal to the waiting time of passenger i, if we define(a) With N equal to the number of passengers that get on the bus, use renewal reward process theory to argue that Image (b)
*35. Satellites are launched according to a Poisson process with rateλ. Each satellite will, independently, orbit the earth for a random time having distribution F. Let denote the number of satellites orbiting at time t.(a) Determine Image.Hint: Relate this to the queue.(b) If at least one
36. Each of n skiers continually, and independently, climbs up and then skis down a particular slope. The time it takes skier i to climb up has distributionI mage, and it is independent of her time to ski down, which has distributionI mage. Let denote the total number of times members of this group
37. There are three machines, all of which are needed for a system to work. Machine i functions for an exponential time with rateI mage before it fails, . When a machine fails, the system is shut down and repair begins on the failed machine. The time to fix machine 1 is exponential with rate 5; the
38. A truck driver regularly drives round trips from A to B and then back to A. Each time he drives from A to B, he drives at a fixed speed that (in miles per hour) is uniformly distributed between 40 and 60; each time he drives from B to A, he drives at a fixed speed that is equally likely to be
39. A system consists of two independent machines that each function for an exponential time with rate λ. There is a single repairperson.If the repairperson is idle when a machine fails, then repair immediately begins on that machine; if the repairperson is busy when a machine fails, then that
40. Three marksmen take turns shooting at a target. Marksman 1 shoots until he misses, then marksman 2 begins shooting until he misses, then marksman 3 until he misses, and then back to marksman 1, and so on. Each time marksman i fires he hits the target, independently of the past, with
41. Consider a waiting line system where customers arrive according to a renewal process, and either enter service if they find a free server or join the queue if all servers are busy. Suppose service times are independent with a distribution H. If we say that an event occurs whenever a departure
42. Dry and wet seasons alternate, with each dry season lasting an exponential time with rate λ and each wet season an exponential time with rate μ. The lengths of dry and wet seasons are all independent. In addition, suppose that people arrive to a service facility according to a Poisson process
43. Individuals arrive two at a time to a 2 server queueing station,with the pairs arriving at times distributed according to a Poisson process with rate λ. A pair will only enter the system if it finds both servers are free. In that case, one member of the pair enters service with server 1 and
44. Consider a renewal reward process where is the nth interarrival time, and where is the reward earned during the nth renewal interval.(a) Give an interpretation of the random variable Image.(b) Find the average value of Image. That is, find Image.
45. Each time a certain machine breaks down it is replaced by a new one of the same type. In the long run, what percentage of time is the machine in use less than one year old if the life distribution of a machine is(a) uniformly distributed over (0, 2)?(b) exponentially distributed with mean 1?
*46. For an interarrival distribution F having mean μ, we defined the equilibrium distribution of F, denoted , by(a) Show that if F is an exponential distribution, then Image.(b) If for some constantc, Image show that is the uniform distribution on (0, c). That is, if interarrival times are
47. Consider a renewal process having interarrival distribution F such that That is, interarrivals are equally likely to be exponential with mean 1 or exponential with mean 2.(a) Without any calculations, guess the equilibrium distribution .(b) Verify your guess in part (a).
*2. Machines in a factory break down at an exponential rate of six per hour. There is a single repairman who fixes machines at an exponential rate of eight per hour. The cost incurred in lost production when machines are out of service is $10 per hour per machine. What is the average cost rate
31. Consider a single-server exponential system in which ordinary customers arrive at a rate λ and have service rate μ. In addition, there is a special customer who has a service rateI mage. Whenever this special customer arrives, she goes directly into service (if anyone else is in service, then
19. Consider a sequential-service system consisting of two servers, A and B. Arriving customers will enter this system only if server A is free. If a customer does enter, then he is immediately served by server A. When his service by A is completed, he then goes to B if B is free, or if B is busy,
20. Customers arrive at a two-server system according to a Poisson process having rateI mage. An arrival finding server 1 free will begin service with that server. An arrival finding server 1 busy and server 2 free will enter service with server 2. An arrival finding both servers busy goes away.
21. Customers arrive at a two-server station in accordance with a Poisson process with a rate of two per hour. Arrivals finding server 1 free begin service with that server. Arrivals finding server 1 busy and server 2 free begin service with server 2. Arrivals finding both servers busy are lost.
22. Arrivals to a three-server system are according to a Poisson process with rate λ. Arrivals finding server 1 free enter service with 1. Arrivals finding 1 busy but 2 free enter service with 2. Arrivals finding both 1 and 2 busy do not join the system. After completion of service at either 1 or
23. The economy alternates between good and bad periods. During good times customers arrive at a certain single-server queueing system in accordance with a Poisson process with rate , and during bad times they arrive in accordance with a Poisson process with rate . A good time period lasts for an
24. There are two types of customers. Type 1 and 2 customers arrive in accordance with independent Poisson processes with respective rate and . There are two servers. A type 1 arrival will enter service with server 1 if that server is free; if server 1 is busy and server 2 is free, then the type 1
*25. Suppose in Exercise 24 we want to find out the proportion of time there is a type 1 customer with server 2. In terms of the longrun probabilities given in Exercise 24, what is(a) the rate at which a type 1 customer enters service with server 2?(b) the rate at which a type 2 customer enters
26. Customers arrive at a single-server station in accordance with a Poisson process with rate λ. All arrivals that find the server free immediately enter service. All service times are exponentially distributed with rate μ. An arrival that finds the server busy will leave the system and roam
27. Consider the system in which customers arrive at rate λand the server serves at rate μ. However, suppose that in any interval of length h in which the server is busy there is a probability Image that the server will experience a breakdown, which causes the system to shut down. All customers
*28. Reconsider Exercise 27, but this time suppose that a customer that is in the system when a breakdown occurs remains there while the server is being fixed. In addition, suppose that new arrivals during a breakdown period are allowed to enter the system. What is the average time a customer
29. Poisson (λ) arrivals join a queue in front of two parallel servers A and B, having exponential service ratesI mage andI mage (see Fig.8.4). When the system is empty, arrivals go into server A with probability α and into B with probability . Otherwise, the head of the queue takes the first
30. In a queue with unlimited waiting space, arrivals are Poisson(parameter λ) and service times are exponentially distributed(parameter μ). However, the server waits until K people are present before beginning service on the first customer; thereafter, he services one at a time until all K
18. Consider a queueing system having two servers and no queue.There are two types of customers. Type 1 customers arrive according to a Poisson process having rate , and will enter the system if either server is free. The service time of a type 1 customer is exponential with rateI mage. Type 2
17. Two customers move about among three servers. Upon completion of service at server i, the customer leaves that server and enters service at whichever of the other two servers is free.(Therefore, there are always two busy servers.) If the service times at server i are exponential with rateI
16. Consider a 2 server system where customers arrive according to a Poisson process with rate λ, and where each arrival is sent to the server currently having the shortest queue. (If they have the same length queue then the choice is made at random.) The service time at either server is
3. The manager of a market can hire either Mary or Alice. Mary, who gives service at an exponential rate of 20 customers per hour, can be hired at a rate of $3 per hour. Alice, who gives service at an exponential rate of 30 customers per hour, can be hired at a rate of$C per hour. The manager
4. In the system, derive by equating the rate at which customers arrive with the rate at which they depart.
5. Suppose customers arrive to a two server system according to a Poisson process with rate λ, and suppose that each arrival is, independently, sent either to server 1 with probability α or to server 2 with probability . Suppose the service time at server i is exponential with rateI mage, .(a)
6. Suppose that a customer of the system spends the amount of timeI mage waiting in queue before entering service.(a) Show that, conditional on the preceding, the number of other customers that were in the system when the customer arrived is distributed as Image, where P is a Poisson random
7. It follows from Exercise 6 that if, in the model, is the amount of time that a customer spends waiting in queue, then Image whereI mage is an exponential random variable with rate . Using this, findI mage.
*8. Show that W is smaller in an model having arrivals at rateλ and service at rate 2μ than it is in a two-server model with arrivals at rate λ and with each server at rate μ. Can you give an intuitive explanation for this result? Would it also be true for ?
9. Consider the queue with impatient customers model as presented in Example 8.9. Give your answers in terms of the limiting probabilities .(a) What is the average amount of time that a customer spends in queue.(b) If Image denotes the probability that a customer who finds n others in the system
10. A facility produces items according to a Poisson process with rateλ. However, it has shelf space for only k items and so it shuts down production whenever k items are present. Customers arrive at the facility according to a Poisson process with rate μ. Each customer wants one item and will
11. A group of n customers moves around among two servers. Upon completion of service, the served customer then joins the queue (or enters service if the server is free) at the other server. All service times are exponential with rate μ. Find the proportion of time that there are j customers at
12. A group of m customers frequents a single-server station in the following manner. When a customer arrives, he or she either enters service if the server is free or joins the queue otherwise. Upon completing service the customer departs the system, but then returns after an exponential time with
*13. Families arrive at a taxi stand according to a Poisson process with rate λ. An arriving family finding N other families waiting for a taxi does not wait. Taxis arrive at the taxi stand according to a Poisson process with rate μ. A taxi finding M other taxis waiting does not wait. Derive
14. Customers arrive to a single server system in accordance with a Poisson process with rate λ. Arrivals only enter if the server is free.Each customer is either a type 1 customer with probability p or a type 2 customer with probability . The time it takes to serve a type i customer is
15. Customers arrive to a two server system in accordance with a Poisson process with rate λ. Server 1 is the preferred server, and an arrival finding server 1 free enters service with 1; an arrival finding 1 busy but 2 free, enters service with 2. Arrivals finding both servers busy do not enter.
The National Health Statistics Reports dated Oct. 22, 2008, stated that for a sample size of 277 18-year-old American males, the sample mean waist circumference was 86.3 cm. A somewhat complicated method was used to estimate various population percentiles, resulting in the following values: 5th
Refer to Exercise 46.Suppose the distribution of diameter is normal.a. Calculate P(11.99 X 12.01) when n = 16.b. How likely is it that the sample mean diameter exceeds 12.01 when n = 25?
The inside diameter of a randomly selected piston ring is a random variable with mean value 12 cm and standard deviation .04 cm.a. If X is the sample mean diameter for a random sample of n = 16 rings, where is the sampling distribution of X centered, and what is the standard deviation of the X
44. Carry out a simulation experiment using a statistical com- puter package or other software to study the sampling dis- tribution of X when the population distribution is Weibull with a 2 and =5, as in Example 5.19. Consider the four sample sizes n = 5, 10, 20, and 30, and in each case use 1000
A company maintains three offices in a certain region, each staffed by two employees. Information concerning yearly salaries (1000s of dollars) is as follows: Office Employee 1 Salary 1 1 2 2 3 3 2 3 4 5 6 29.7 33.6 30.2 33.6 25.8 29.7a. Suppose two of these employees are randomly selected from
Let X be the number of packages being mailed by a ran- domly selected customer at a certain shipping facility. Suppose the distribution of X is as follows: x p(x) 1 2 3 4 4 3 2 1a. Consider a random sample of size n = 2 (two cus- tomers), and let X be the sample mean number of pack- ages shipped.
It is known that 80% of all brand A zip drives work in a sat- isfactory manner throughout the warranty period (are "suc- cesses"). Suppose that n = 10 drives are randomly selected. Let X = the number of successes in the sample. The statistic Xn is the sample proportion (fraction) of successes.
There are two traffic lights on a commuter's route to and from work. Let X, be the number of lights at which the com- muter must stop on his way to work, and X, be the number of lights at which he must stop when returning from work. Suppose these two variables are independent, each with pmf given
The breaking strength of a rivet has a mean value of 10,000 psi and a standard deviation of 500 psi.a. What is the probability that the sample mean breaking strength for a random sample of 40 rivets is between 9900 and 10,200?b. If the sample size had been 15 rather than 40, could the probability
If two loads are applied to a cantilever beam as shown in the accompanying drawing, the bending moment at 0 due to the loads is aX+aX. (0, 1)- (x, 1-x)a. Suppose that X, and X, are independent rv's with means 2 and 4 kips, respectively, and standard deviations .5 and 1.0 kip, respectively. Ifa, 5
Suppose that when the pH of a certain chemical compound is 5.00, the pH measured by a randomly selected begin- ning chemistry student is a random variable with mean 5.00 and standard deviation .2. A large batch of the com- pound is subdivided and a sample given to each student in a morning lab and
Suppose your waiting time for a bus in the morning is uni- formly distributed on [0, 8], whereas waiting time in the evening is uniformly distributed on [0, 10] independent of morning waiting time.a. If you take the bus each morning and evening for a week. what is your total expected waiting time?
Refer to Exercise 3.a. Calculate the covariance between X = the number of customers in the express checkout and X = the number of customers in the superexpress checkout.b. Calculate V(X + X2). How does this compare to V(X)) + V(X2)?
Five automobiles of the same type are to be driven on a 300- mile trip. The first two will use an economy brand of gaso- line, and the other three will use a name brand. Let X1, X2, X3, X4, and X; be the observed fuel efficiencies (mpg) for the five cars. Suppose these variables are independent and
Let X1, X2, and X3 represent the times necessary to perform three successive repair tasks at a certain service facility. Suppose they are independent, normal rv's with expected values , , and , and variances , , and , respec- tively.a. If = 60 and = = = 15, calculate == P(T200) and P(150 T =
A shipping company handles containers in three different sizes: (1) 27 ft (3 x 3 x 3), (2) 125 ft', and (3) 512 ft. Let X, (i = 1, 2, 3) denote the number of type i containers shipped during a given week. With , = E(X) and =V(X), suppose that the mean values and standard deviations are as follows:
The number of parking tickets issued in a certain city on any given weekday has a Poisson distribution with parameter =50. What is the approximate probability thata. Between 35 and 70 tickets are given out on a particular day? [Hint: When is large, a Poisson rv has approxi- mately a normal
a. Recalling the definition of o for a single rv X, write a formula that would be appropriate for computing the variance of a function h(X, Y) of two random variables. [Hint: Remember that variance is just a special expected value.]b. Use this formula to compute the variance of the recorded score
Consider a system consisting of three components as pic- tured. The system will continue to function as long as the first component functions and either component 2 or com- ponent 3 functions. Let X, X, and X, denote the lifetimes of components 1, 2, and 3, respectively. Suppose the X's are
Two components of a minicomputer have the following joint pdf for their useful lifetimes X and Y: (xex(1+y) x = 0 and y=0 f(x, y) 0 otherwisea. What is the probability that the lifetime X of the first component exceeds 3?b. What are the marginal pdf's of X and Y? Are the two life- times
Two different professors have just submitted final exams for duplication. Let X denote the number of typographical errors on the first professor's exam and Y denote the number of such errors on the second exam. Suppose X has a Poisson distribution with parameter p, Y has a Poisson distribution with
Annie and Alvie have agreed to meet between 5:00 P.M. and 6:00 P.M. for dinner at a local health-food restaurant. Let X = Annie's arrival time and Y=Alvie's arrival time. Suppose X and Y are independent with each uniformly dis- tributed on the interval [5, 6].a. What is the joint pdf of X and Y?b.
Each front tire on a particular type of vehicle is supposed to be filled to a pressure of 26 psi. Suppose the actual air pres- sure in each tire is a random variable-X for the right tire and Y for the left tire, with joint pdf F(x,y) = { K(x + JK(x + y) 20 x 30, 20 y 30 otherwise 0a. What is the
A stockroom currently has 30 components of a certain type, of which 8 were provided by supplier 1, 10 by supplier 2.and 12 by supplier 3.Six of these are to be randomly selected for a particular assembly. Let X = the number of supplier 1's components selected, Y = the number of sup- plier 2's
The joint probability distribution of the number X of cars and the number Y of buses per signal cycle at a proposed left-turn lane is displayed in the accompanying joint probability table. p(x, y) x 012345 y 0 1 2 .025 .015 .010 .050 .030 .020 .125 .075 .050 .150 .090 .060 .100 .060 .040 .050 .030
Let X denote the number of Canon digital cameras sold dur- ing a particular week by a certain store. The pmf of X is x 0 1 2 3 4 Px(x) .1 .2 .3 .25 .15 Sixty percent of all customers who purchase these cameras also buy an extended warranty. Let Y denote the number of purchasers during this week who
Return to the situation described in Exercise 3.a. Determine the marginal pmf of X, and then calculate the expected number of customers in line at the express checkout.b. Determine the marginal pmf of X.c. By inspection of the probabilities P(X = 4), P(X = 0), and P(X = 4, X=0), are X, and X,
Showing 2100 - 2200
of 7136
First
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
Last
Step by Step Answers
Get 50% OFF
Claim Now
