Banner Ad 1

Collapse

Announcement

Collapse
No announcement yet.

Transformations problem

Collapse
This topic is closed.
X
X
 
  • Filter
  • Time
  • Show
Clear All
new posts

  • Transformations problem

    Can anybody try to solve and explain this problem?

    X and y have a joint distribution with pdf f(x,y)=e^ -(x+y), x>0 y>0. The random variable Z is defined to be equal to U=e^ -(x+y). Find the pdf of f (u).

    Answer: f(u)= -ln u for 0<u<1

    Thanks

  • #2
    Transformation problem

    Originally posted by Rup
    Can anybody try to solve and explain this problem?
    X and y have a joint distribution with pdf f(x,y)=e^ -(x+y), x>0 y>0. The random variable Z is defined to be equal to U=e^ -(x+y). Find the pdf of f (u).
    Answer: f(u)= -ln u for 0<u<1
    Thanks
    You know immediately that W = X + Y has gamma distribution with parameters alpha = 2, beta = 1, and PDF w*exp(-w) for w > 0 (if you do not know this immediately, revise your study strategy, and do not use minimalist approach -- you are supposed to know this, and your study material should have told you that). Then it is a simple one variable transformation problem Z = exp (-W).

    The serious questions for you are:
    - You should know how to quickly find the distribution of X + Y for any random variables X and Y, do you? The answer is: convolution.
    - You should know what a sum of two IID exponentials is, do you? The answer is: gamma distribution.
    - You should know how to quickly find a PDF of a transformation of a single random variable, and a two-component random vector, do you?
    Make certain that you study these. Most of this is addressed in a single, complicated but didactic, exercise I posted on May 14, 2005 on this forum.

    I will post your question as an exercise. But the above solution, in first lines, is nearly immediate.

    Yours,
    Krzys'
    Want to know how to pass actuarial exams? Go to: smartURL.it/pass

    Comment

    Working...
    X