An exponentially distributed random variable

1. The time T required to repair a machine is an exponentially distributed random
variable with mean 1
2 (hours).
(a) What is the probability that a repair time exceeds 1
2 hour?
(b) What is the probability that a repair takes at least 121
2 hours given that
its duration exceeds 12 hours?
2. Suppose that you arrive at a single-teller bank to find five other customers in
the bank, one being served and the other four waiting in line. You join the
end of the line. If the service times are all exponential with rate μ, what is the
expected amount of time you will spend in the bank?
3. Let X be an exponential random variable. Without any computations, tell
which one of the following is correct. Explain your answer.
(a) E[X2|X>1] = E[(X +1)2] (b) E[X2|X>1] = E[X2] + 1
(c) E[X2|X>1] = (1 +E[X])2
4. Consider a post office with two clerks. Three people, A, B, and C, enter simultaneously.
A and B go directly to the clerks, and C waits until either A or
B leaves before he begins service. What is the probability that A is still in the
post office after the other two have left when
(a) the service time for each clerk is exactly (nonrandom) ten minutes?
(b) the service times are i with probability 1
3 , i = 1, 2, 3?
(c) the service times are exponential with mean 1/μ?
*5. If X is exponential with rate !, show that Y = [X] + 1 is geometric with parameter
p = 1 −e−!, where [x] is the largest integer less than or equal to x.
6. In Example 5.3 if server i serves at an exponential rate !i, i = 1, 2, show that
P{Smith is not last} =
!1 +!2
!1 + !2
*7. If X1 and X2 are independent nonnegative continuous random variables, show
P{X1 <X2|min(X1,X2) = t} =
r1(t) +r2(t)
where ri (t) is the failure rate function of Xi .
8. If X and Y are independent exponential random variables with respective rates
! and μ, what is the conditional distribution of X given thatX<Y?
9. Machine 1 is currently working. Machine 2 will be put in use at a time t from
now. If the lifetime of machine i is exponential with rate !i, i = 1, 2, what is
the probability that machine 1 is the first machine to fail?
*10. Let X and Y be independent exponential random variables with respective
rates ! and μ. Let M = min(X, Y). Find
(a) E[MX|M = X],
358 Introduction to Probability Models
(b) E[MX|M = Y],
(c) Cov(X,M).
11. Let X, Y1, . . . , Yn be independent exponential random variables; X having rate
!, and Yi having rate μ. Let Aj be the event that the j th smallest of these
n + 1 random variables is one of the Yi. Find p = P{X >maxi Yi }, by using
the identity
p = P(A1 ···An) = P(A1)P (A2|A1) ···P(An|A1 ···An−1)
Verify your answer when n = 2 by conditioning on X to obtain p.
12. If Xi, i = 1, 2, 3, are independent exponential random variables with rates !i ,
i = 1, 2, 3, find
(a) P{X1 <X2 <X3},
(b) P{X1 <X2|max(X1, X2, X3) = X3},
(c) E[maxXi |X1 <X2 <X3],
(d) E[maxXi ].
13. Find, in Example 5.10, the expected time until the nth person on line leaves
the line (either by entering service or departing without service).
14. I am waiting for two friends to arrive at my house. The time until A arrives is
exponentially distributed with rate !a, and the time until B arrives is exponentially
distributed with rate !b. Once they arrive, both will spend exponentially
distributed times, with respective rates μa and μb at my home before departing.
The four exponential random variables are independent.
(a) What is the probability that A arrives before and departs after B?
(b) What is the expected time of the last departure?
15. One hundred items are simultaneously put on a life test. Suppose the lifetimes
of the individual items are independent exponential random variables
with mean 200 hours. The test will end when there have been a total of 5
failures. If T is the time at which the test ends, find E[T ] and Var(T ).
16. There are three jobs that need to be processed, with the processing time of
job i being exponential with rate μi . There are two processors available, so
processing on two of the jobs can immediately start, with processing on the
final job to start when one of the initial ones is finished.
(a) Let Ti denote the time at which the processing of job i is completed.
If the objective is to minimize E[T1 + T2 + T3], which jobs should be
initially processed if μ1 <μ2 <μ3?
(b) Let M, called the makespan, be the time until all three jobs have been
processed. With S equal to the time that there is only a single processor
working, show that
2E[M] = E[S] +
For the rest of this problem, suppose that μ1 = μ2 = μ, μ3 = !. Also,
let P(μ) be the probability that the last job to finish is either job 1 or job
The Exponential Distribution and the Poisson Process 359
2, and let P(!) = 1 − P(μ) be the probability that the last job to finish
is job 3.
(c) Express E[S] in terms of P(μ) and P(!).
Let Pi,j (μ) be the value of P(μ) when i and j are the jobs that are
initially started.
(d) Show that P1,2(μ) ” P1,3(μ).
(e) If μ>! show that E[M] is minimized when job 3 is one of the jobs that
is initially started.
(f) If μ < ! show that E[M] is minimized when processing is initially
started on jobs 1 and 2.
17. A set of n cities is to be connected via communication links. The cost to
construct a link between cities i and j is Cij , i #= j. Enough links should be
constructed so that for each pair of cities there is a path of links that connects
them. As a result, only n − 1 links need be constructed. A minimal cost algorithm
for solving this problem (known as the minimal spanning tree problem)
first constructs the cheapest of all the
links. Then, at each additional stage
it chooses the cheapest link that connects a city without any links to one with
links. That is, if the first link is between cities 1 and 2, then the second link
will either be between 1 and one of the links 3, . . . , n or between 2 and one of
the links 3, . . . , n. Suppose that all of the
costs Cij are independent exponential
random variables with mean 1. Find the expected cost of the preceding
algorithm if
(a) n = 3,
(b) n = 4.
*18. Let X1 and X2 be independent exponential random variables, each having rate
μ. Let
X(1) = minimum(X1,X2) and X(2) = maximum(X1,X2)
(a) E[X(1)],
(b) Var[X(1)],
(c) E[X(2)],
(d) Var[X(2)].
19. In a mile race between A and B, the time it takes A to complete the mile is
an exponential random variable with rate !a and is independent of the time it
takes B to complete the mile, which is an exponential random variable with
rate !b. The one who finishes earliest is declared the winner and receives
Re−”t if the winning time is t, where R and ” are constants. If the loser
receives 0, find the expected amount that runner A wins.
20. Consider a two-server system in which a customer is served first by server 1,
then by server 2, and then departs. The service times at server i are exponential
random variables with rates μi, i = 1, 2. When you arrive, you find server 1
free and two customers at server 2—customer A in service and customer B
waiting in line.
360 Introduction to Probability Models
(a) Find PA, the probability that A is still in service when you move over to
server 2.
(b) Find PB, the probability that B is still in the system when you move over
to server 2.
(c) Find E[T ], where T is the time that you spend in the system.
Hint: Write
T = S1 +S2 +WA +WB
where Si is your service time at server i,WA is the amount of time you wait
in queue while A is being served, and WB is the amount of time you wait in
queue while B is being served.
21. In a certain system, a customer must first be served by server 1 and then by
server 2. The service times at server i are exponential with rate μi, i = 1, 2.
An arrival finding server 1 busy waits in line for that server. Upon completion
of service at server 1, a customer either enters service with server 2 if that
server is free or else remains with server 1 (blocking any other customer from
entering service) until server 2 is free. Customers depart the system after being
served by server 2. Suppose that when you arrive there is one customer in the
system and that customer is being served by server 1. What is the expected
total time you spend in the system?
22. Suppose in Exercise 21 you arrive to find two others in the system, one being
served by server 1 and one by server 2. What is the expected time you spend
in the system? Recall that if server 1 finishes before server 2, then server 1’s
customer will remain with him (thus blocking your entrance) until server 2
becomes free.
*23. A flashlight needs two batteries to be operational. Consider such a flashlight
along with a set of n functional batteries—battery 1, battery 2, . . . , battery n.
Initially, battery 1 and 2 are installed. Whenever a battery fails, it is immediately
replaced by the lowest numbered functional battery that has not yet been
put in use. Suppose that the lifetimes of the different batteries are independent
exponential random variables each having rate μ. At a randomtime, call it T ,
a battery will fail and our stockpilewill be empty.At that moment exactly one
of the batteries—which we call battery X—will not yet have failed.
(a) What is P{X = n}?
(b) What is P{X = 1}?
(c) What is P{X = i}?
(d) Find E[T ].
(e) What is the distribution of T ?
24. There are two servers available to process n jobs. Initially, each server begins
work on a job. Whenever a server completes work on a job, that job leaves
the system and the server begins processing a new job (provided there are still
jobs waiting to be processed). Let T denote the time until all jobs have been
processed. If the time that it takes server i to process a job is exponentially
distributed with rate μi, i = 1, 2, find E[T ] and Var(T ).

Get Professional Assignment Help Cheaply

Buy Custom Essay

Don't use plagiarized sources. Get Your Custom Essay on
An exponentially distributed random variable
Just from $9/Page
Order Essay

Are you busy and do not have time to handle your assignment? Are you scared that your paper will not make the grade? Do you have responsibilities that may hinder you from turning in your assignment on time? Are you tired and can barely handle your assignment? Are your grades inconsistent?

Whichever your reason is, it is valid! You can get professional academic help from our service at affordable rates. We have a team of professional academic writers who can handle all your assignments.

Why Choose Our Academic Writing Service?

  • Plagiarism free papers
  • Timely delivery
  • Any deadline
  • Skilled, Experienced Native English Writers
  • Subject-relevant academic writer
  • Adherence to paper instructions
  • Ability to tackle bulk assignments
  • Reasonable prices
  • 24/7 Customer Support
  • Get superb grades consistently

Online Academic Help With Different Subjects


Students barely have time to read. We got you! Have your literature essay or book review written without having the hassle of reading the book. You can get your literature paper custom-written for you by our literature specialists.


Do you struggle with finance? No need to torture yourself if finance is not your cup of tea. You can order your finance paper from our academic writing service and get 100% original work from competent finance experts.

Computer science

Computer science is a tough subject. Fortunately, our computer science experts are up to the match. No need to stress and have sleepless nights. Our academic writers will tackle all your computer science assignments and deliver them on time. Let us handle all your python, java, ruby, JavaScript, php , C+ assignments!


While psychology may be an interesting subject, you may lack sufficient time to handle your assignments. Don’t despair; by using our academic writing service, you can be assured of perfect grades. Moreover, your grades will be consistent.


Engineering is quite a demanding subject. Students face a lot of pressure and barely have enough time to do what they love to do. Our academic writing service got you covered! Our engineering specialists follow the paper instructions and ensure timely delivery of the paper.


In the nursing course, you may have difficulties with literature reviews, annotated bibliographies, critical essays, and other assignments. Our nursing assignment writers will offer you professional nursing paper help at low prices.


Truth be told, sociology papers can be quite exhausting. Our academic writing service relieves you of fatigue, pressure, and stress. You can relax and have peace of mind as our academic writers handle your sociology assignment.


We take pride in having some of the best business writers in the industry. Our business writers have a lot of experience in the field. They are reliable, and you can be assured of a high-grade paper. They are able to handle business papers of any subject, length, deadline, and difficulty!


We boast of having some of the most experienced statistics experts in the industry. Our statistics experts have diverse skills, expertise, and knowledge to handle any kind of assignment. They have access to all kinds of software to get your assignment done.


Writing a law essay may prove to be an insurmountable obstacle, especially when you need to know the peculiarities of the legislative framework. Take advantage of our top-notch law specialists and get superb grades and 100% satisfaction.

What discipline/subjects do you deal in?

We have highlighted some of the most popular subjects we handle above. Those are just a tip of the iceberg. We deal in all academic disciplines since our writers are as diverse. They have been drawn from across all disciplines, and orders are assigned to those writers believed to be the best in the field. In a nutshell, there is no task we cannot handle; all you need to do is place your order with us. As long as your instructions are clear, just trust we shall deliver irrespective of the discipline.

Are your writers competent enough to handle my paper?

Our essay writers are graduates with bachelor's, masters, Ph.D., and doctorate degrees in various subjects. The minimum requirement to be an essay writer with our essay writing service is to have a college degree. All our academic writers have a minimum of two years of academic writing. We have a stringent recruitment process to ensure that we get only the most competent essay writers in the industry. We also ensure that the writers are handsomely compensated for their value. The majority of our writers are native English speakers. As such, the fluency of language and grammar is impeccable.

What if I don’t like the paper?

There is a very low likelihood that you won’t like the paper.

Reasons being:

  • When assigning your order, we match the paper’s discipline with the writer’s field/specialization. Since all our writers are graduates, we match the paper’s subject with the field the writer studied. For instance, if it’s a nursing paper, only a nursing graduate and writer will handle it. Furthermore, all our writers have academic writing experience and top-notch research skills.
  • We have a quality assurance that reviews the paper before it gets to you. As such, we ensure that you get a paper that meets the required standard and will most definitely make the grade.

In the event that you don’t like your paper:

  • The writer will revise the paper up to your pleasing. You have unlimited revisions. You simply need to highlight what specifically you don’t like about the paper, and the writer will make the amendments. The paper will be revised until you are satisfied. Revisions are free of charge
  • We will have a different writer write the paper from scratch.
  • Last resort, if the above does not work, we will refund your money.

Will the professor find out I didn’t write the paper myself?

Not at all. All papers are written from scratch. There is no way your tutor or instructor will realize that you did not write the paper yourself. In fact, we recommend using our assignment help services for consistent results.

What if the paper is plagiarized?

We check all papers for plagiarism before we submit them. We use powerful plagiarism checking software such as SafeAssign, LopesWrite, and Turnitin. We also upload the plagiarism report so that you can review it. We understand that plagiarism is academic suicide. We would not take the risk of submitting plagiarized work and jeopardize your academic journey. Furthermore, we do not sell or use prewritten papers, and each paper is written from scratch.

When will I get my paper?

You determine when you get the paper by setting the deadline when placing the order. All papers are delivered within the deadline. We are well aware that we operate in a time-sensitive industry. As such, we have laid out strategies to ensure that the client receives the paper on time and they never miss the deadline. We understand that papers that are submitted late have some points deducted. We do not want you to miss any points due to late submission. We work on beating deadlines by huge margins in order to ensure that you have ample time to review the paper before you submit it.

Will anyone find out that I used your services?

We have a privacy and confidentiality policy that guides our work. We NEVER share any customer information with third parties. Noone will ever know that you used our assignment help services. It’s only between you and us. We are bound by our policies to protect the customer’s identity and information. All your information, such as your names, phone number, email, order information, and so on, are protected. We have robust security systems that ensure that your data is protected. Hacking our systems is close to impossible, and it has never happened.

How our Assignment Help Service Works

1. Place an order

You fill all the paper instructions in the order form. Make sure you include all the helpful materials so that our academic writers can deliver the perfect paper. It will also help to eliminate unnecessary revisions.

2. Pay for the order

Proceed to pay for the paper so that it can be assigned to one of our expert academic writers. The paper subject is matched with the writer’s area of specialization.

3. Track the progress

You communicate with the writer and know about the progress of the paper. The client can ask the writer for drafts of the paper. The client can upload extra material and include additional instructions from the lecturer. Receive a paper.

4. Download the paper

The paper is sent to your email and uploaded to your personal account. You also get a plagiarism report attached to your paper.

smile and order essay GET A PERFECT SCORE!!! smile and order essay Buy Custom Essay

Place your order
(550 words)

Approximate price: $22

Calculate the price of your order

550 words
We'll send you the first draft for approval by September 11, 2018 at 10:52 AM
Total price:
The price is based on these factors:
Academic level
Number of pages
Basic features
  • Free title page and bibliography
  • Unlimited revisions
  • Plagiarism-free guarantee
  • Money-back guarantee
  • 24/7 support
On-demand options
  • Writer’s samples
  • Part-by-part delivery
  • Overnight delivery
  • Copies of used sources
  • Expert Proofreading
Paper format
  • 275 words per page
  • 12 pt Arial/Times New Roman
  • Double line spacing
  • Any citation style (APA, MLA, Chicago/Turabian, Harvard)

Our guarantees

Delivering a high-quality product at a reasonable price is not enough anymore.
That’s why we have developed 5 beneficial guarantees that will make your experience with our service enjoyable, easy, and safe.

Money-back guarantee

You have to be 100% sure of the quality of your product to give a money-back guarantee. This describes us perfectly. Make sure that this guarantee is totally transparent.

Read more

Zero-plagiarism guarantee

Each paper is composed from scratch, according to your instructions. It is then checked by our plagiarism-detection software. There is no gap where plagiarism could squeeze in.

Read more

Free-revision policy

Thanks to our free revisions, there is no way for you to be unsatisfied. We will work on your paper until you are completely happy with the result.

Read more

Privacy policy

Your email is safe, as we store it according to international data protection rules. Your bank details are secure, as we use only reliable payment systems.

Read more

Fair-cooperation guarantee

By sending us your money, you buy the service we provide. Check out our terms and conditions if you prefer business talks to be laid out in official language.

Read more
error: Content is protected !!
Open chat
Need assignment help? You can contact our live agent via WhatsApp using +1 718 717 2861

Feel free to ask questions, clarifications, or discounts available when placing an order.
  +1 718 717 2861           + 44 161 818 7126           [email protected]
  +1 718 717 2861         [email protected]