Answer in Statistics and Probability for pas #252321

Suppose the joint p.d.f of the continuous random variable X and Y is given by

f(x,y)=λ2e(-λ(x+y)) zero otherwise x>0 y>0

Don't use plagiarized sources. Get Your Custom Essay on
Answer in Statistics and Probability for pas #252321
Just from $13/Page
Order Essay

i) Obtain the joint p.d.f of the new random variables V=X+Y and U=X/(X+Y)

ii)Find the marginal probability distribution functions of U and V

iii) Determine whether or not U and V are independent

“f(x,y)=\lambda^2e^{-\lambda(x+y)}, \space x,y\gt0”

The new random variables are defined by,

“V=X+Y” and “U=X/(X+Y)”


To find the joint probability distribution functions for these random variables, we shall apply the transformation of continuous random variable using “Jacobian” transformation as described below.

Let us first make the random variables “X” and “Y” subject of the formula then,

“X=UV” and “Y=V(1-U)”. We define “Jacobian(J)” as a “2\times2” matrix given as,

We now apply the Jacobian transformation as follows,

“J=\begin{vmatrix}n dx/du & dx/dv \\n dy/du & dy/dvn\end{vmatrix}=\begin{vmatrix}n V & U \\n -V & 1-Un\end{vmatrix}=(V*(1-U))+UV=V”

The joint probability distribution function of “U” and “V” is given as,


Therefore, the joint pdf for “U” and V is given as,

“g(u,v)=\lambda^2ve^{-\lambda v},\space 0\lt u\lt1,\space v\gt 0”

“0, \space elsewhere”

“ii)” The marginal distributions of the random variables are obtained as described below.

Given that the joint pdf is given by,

“g(u,v)=\lambda^2ve^{-\lambda v}, \space 0\lt u\lt 1, v\gt0” ,

Then the marginal distribution of the random variable “V” is given as,


“h(v)=\int^1_0(\lambda^2ve^{-\lambda v})du”

“h(v)=(\lambda^2ve^{-\lambda v})*u|^1_0=\lambda^2ve^{-\lambda v}”


“h(v)=\lambda^2ve^{-\lambda v},\space v\gt0”

“0,\space elsewhere”

and the marginal distribution of the random variable “U”is given as,


“f(u)=\int^\infin_0\lambda^2ve^{-\lambda v}dv”

Let us evaluate the integral first,

“\int^\infin_0\lambda^2ve^{-\lambda v}dv=\lambda^2\int^\infin_0ve^{-\lambda v}dv”

This integral can be solved using integration by parts method as below,


“x=v” and “dy/dv=e^{-\lambda v}”

we need to find “dx/dv” and “y”

“dx/dv=1” and “y=\int e^{-\lambda v }dv=-(1/\lambda)e^{-\lambda v}”

Thus the integral can be written as,

“\lambda^2\int^\infin_0ve^{-\lambda v}dv=\lambda^2((-v/\lambda)e^{-\lambda v}|^\infin_0+(1/\lambda)\int^\infin_0 e^{-\lambda v}dv)”

“=\lambda^2(-(v/\lambda)e^{-\lambda v}-(1/\lambda^2)e^{-\lambda v})|^\infin_0”


Therefore the marginal distribution for the random variable “U” given as,

“f(u)=1,\space 0\lt u\lt 1”

“0, \space elsewhere”


If the random variables “U”and “V” are independent then we expect that their joint probability distribution function“g(u,v)” must equal the product of their marginal distribution functions as written below.

If “U” and “V”are independent then,



“\lambda^2ve^{-\lambda v}=1*\lambda^2ve^{-\lambda v}”

Since the product of the marginal distributions is equal to the joint distribution then, “U” and “V” are independent.

What Will You Get?

We provide professional writing services to help you score straight A’s by submitting custom written assignments that mirror your guidelines.

Premium Quality

Get result-oriented writing and never worry about grades anymore. We follow the highest quality standards to make sure that you get perfect assignments.

Experienced Writers

Our writers have experience in dealing with papers of every educational level. You can surely rely on the expertise of our qualified professionals.

On-Time Delivery

Your deadline is our threshold for success and we take it very seriously. We make sure you receive your papers before your predefined time.

24/7 Customer Support

Someone from our customer support team is always here to respond to your questions. So, hit us up if you have got any ambiguity or concern.

Complete Confidentiality

Sit back and relax while we help you out with writing your papers. We have an ultimate policy for keeping your personal and order-related details a secret.

Authentic Sources

We assure you that your document will be thoroughly checked for plagiarism and grammatical errors as we use highly authentic and licit sources.

Moneyback Guarantee

Still reluctant about placing an order? Our 100% Moneyback Guarantee backs you up on rare occasions where you aren’t satisfied with the writing.

Order Tracking

You don’t have to wait for an update for hours; you can track the progress of your order any time you want. We share the status after each step.


Areas of Expertise

Although you can leverage our expertise for any writing task, we have a knack for creating flawless papers for the following document types.

Areas of Expertise

Although you can leverage our expertise for any writing task, we have a knack for creating flawless papers for the following document types.


Trusted Partner of 9650+ Students for Writing

From brainstorming your paper's outline to perfecting its grammar, we perform every step carefully to make your paper worthy of A grade.

Preferred Writer

Hire your preferred writer anytime. Simply specify if you want your preferred expert to write your paper and we’ll make that happen.

Grammar Check Report

Get an elaborate and authentic grammar check report with your work to have the grammar goodness sealed in your document.

One Page Summary

You can purchase this feature if you want our writers to sum up your paper in the form of a concise and well-articulated summary.

Plagiarism Report

You don’t have to worry about plagiarism anymore. Get a plagiarism report to certify the uniqueness of your work.

Free Features $66FREE

  • Most Qualified Writer $10FREE
  • Plagiarism Scan Report $10FREE
  • Unlimited Revisions $08FREE
  • Paper Formatting $05FREE
  • Cover Page $05FREE
  • Referencing & Bibliography $10FREE
  • Dedicated User Area $08FREE
  • 24/7 Order Tracking $05FREE
  • Periodic Email Alerts $05FREE

Our Services

Join us for the best experience while seeking writing assistance in your college life. A good grade is all you need to boost up your academic excellence and we are all about it.

  • On-time Delivery
  • 24/7 Order Tracking
  • Access to Authentic Sources
Academic Writing

We create perfect papers according to the guidelines.

Professional Editing

We seamlessly edit out errors from your papers.

Thorough Proofreading

We thoroughly read your final draft to identify errors.


Delegate Your Challenging Writing Tasks to Experienced Professionals

Work with ultimate peace of mind because we ensure that your academic work is our responsibility and your grades are a top concern for us!

Check Out Our Sample Work

Dedication. Quality. Commitment. Punctuality

It May Not Be Much, but It’s Honest Work!

Here is what we have achieved so far. These numbers are evidence that we go the extra mile to make your college journey successful.


Happy Clients


Words Written This Week


Ongoing Orders


Customer Satisfaction Rate

Process as Fine as Brewed Coffee

We have the most intuitive and minimalistic process so that you can easily place an order. Just follow a few steps to unlock success.

See How We Helped 9000+ Students Achieve Success


We Analyze Your Problem and Offer Customized Writing

We understand your guidelines first before delivering any writing service. You can discuss your writing needs and we will have them evaluated by our dedicated team.

  • Clear elicitation of your requirements.
  • Customized writing as per your needs.

We Mirror Your Guidelines to Deliver Quality Services

We write your papers in a standardized way. We complete your work in such a way that it turns out to be a perfect description of your guidelines.

  • Proactive analysis of your writing.
  • Active communication to understand requirements.

We Handle Your Writing Tasks to Ensure Excellent Grades

We promise you excellent grades and academic excellence that you always longed for. Our writers stay in touch with you via email.

  • Thorough research and analysis for every order.
  • Deliverance of reliable writing service to improve your grades.
Place an Order Start Chat Now