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Rup
November 24th 2005, 05:30 PM
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

krzysio
November 25th 2005, 01:42 PM
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'