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introduction to probability statistics

Questions and Answers of Introduction To Probability Statistics

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
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
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,
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
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
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
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
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
*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
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)
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
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
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
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
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
*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
*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
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
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
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
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
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,
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
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
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
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)
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
*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
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
*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
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.
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
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
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
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.
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
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
*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
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
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
*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
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
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
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
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,
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
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
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
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
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,
*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
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
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
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
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
*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
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
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
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
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
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,
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
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
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
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
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
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
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
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.
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
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
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
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
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
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
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,
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
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
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
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
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
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
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
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

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