1. Let X and Y be random variables taking the values on the set 0 ≤ X ≤ Y with joint density function
fX,Y (x, y) = (
for 0 ≤ x ≤ y

0 otherwise.
(a) Draw the support of the joint density function. Are X and Y independent? Why or why not?
(b) Find the marginal density function for Y .
(c) Write a formula for the conditional density of X given that Y = y for an arbitrary value of y,
fX|Y (x|y).
(d) Compute the conditional expectation E(X|Y = y). The result should depend on the value of y
but not x.
(e) Based on your answer to part (d), do you think the covariance of X and Y should be positive,
negative, or zero? Explain.
2. The number of emails I will receive as I sleep tonight is a Poisson random variable with mean 30. Each
time I receive an email, the probability that it is spam is 0.3, independent of all other emails. Thus
if I receive N emails, the number of emails which are spam is a Bin(N, 0.3) random variable. Let N
be the number of emails I will receive tonight, and let X be the number of spam emails I will receive
(a) Use “double expectation” to compute the expected value of the number of spam emails I will
receive tonight, E(X).
(b) Use “double expectation” to compute the variance of the number of spam emails I will receive
tonight, Var(X). You may need to use the fact that E(N2
) = E(N)
(c) Without doing any computations, do you expect the covariance of X and N to be positive,
negative, or zero? Explain why.

Don't use plagiarized sources. Get Your Custom Essay on
Just from $13/Page
Order Essay

(d) Use “double expectation” and your answer to part (a) to compute the covariance of N and X.
Hint: The covariance is E(XN)−E(X)E(N). Since E(N) = 30 and you found E(X) in part (a),
you just need to find E(XN). Conditioning on N, this is E(E(XN|N)) = E(NE(X|N)), since
when conditioning on N, we are thinking of N as a constant. You will have to use the fact that
) = E(N)
(e) Use your answers to parts (b) and (d) to find the correlation coefficient between X and N.
3. A recent NYT/Sienna poll showed that in the upcoming presidential election, 36% of Americans plan
to support President Trump, 50% plan to support the presumptive Democrat nominee, Joe Biden,
and the remaining 14% are undecided. Assume these numbers are correct, and suppose we gather
5 randomly selected Americans in a room. Find the probability that there are the same number of
Trump supporters as Biden supporters in the room. That is, if T is the number of Trump supporters,
and B is the number of Biden supporters, find P(T = B).

Acme Writers
Calculate your paper price
Pages (550 words)
Approximate price: -

Why Work with Us

Top Quality and Well-Researched Papers

We always make sure that writers follow all your instructions precisely. You can choose your academic level: high school, college/university or professional, and we will assign a writer who has a respective degree.

Professional and Experienced Academic Writers

We have a team of professional writers with experience in academic and business writing. Many are native speakers and able to perform any task for which you need help.

Free Unlimited Revisions

If you think we missed something, send your order for a free revision. You have 10 days to submit the order for review after you have received the final document. You can do this yourself after logging into your personal account or by contacting our support.

Prompt Delivery and 100% Money-Back-Guarantee

All papers are always delivered on time. In case we need more time to master your paper, we may contact you regarding the deadline extension. In case you cannot provide us with more time, a 100% refund is guaranteed.

Original & Confidential

We use several writing tools checks to ensure that all documents you receive are free from plagiarism. Our editors carefully review all quotations in the text. We also promise maximum confidentiality in all of our services.

24/7 Customer Support

Our support agents are available 24 hours a day 7 days a week and committed to providing you with the best customer experience. Get in touch whenever you need any assistance.

Try it now!

Calculate the price of your order

Total price:

How it works?

Follow these simple steps to get your paper done

Place your order

Fill in the order form and provide all details of your assignment.

Proceed with the payment

Choose the payment system that suits you most.

Receive the final file

Once your paper is ready, we will email it to you.

Our Services

No need to work on your paper at night. Sleep tight, we will cover your back. We offer all kinds of writing services.


Essay Writing Service

No matter what kind of academic paper you need and how urgent you need it, you are welcome to choose your academic level and the type of your paper at an affordable price. We take care of all your paper needs and give a 24/7 customer care support system.


Admission Essays & Business Writing Help

An admission essay is an essay or other written statement by a candidate, often a potential student enrolling in a college, university, or graduate school. You can be rest assurred that through our service we will write the best admission essay for you.


Editing Support

Our academic writers and editors make the necessary changes to your paper so that it is polished. We also format your document by correctly quoting the sources and creating reference lists in the formats APA, Harvard, MLA, Chicago / Turabian.


Revision Support

If you think your paper could be improved, you can request a review. In this case, your paper will be checked by the writer or assigned to an editor. You can use this option as many times as you see fit. This is free because we want you to be completely satisfied with the service offered.