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Suppose $X$ and $Y$ are independent exponential random variables. So $f(x) = (1/\lambda) e^{-x/\lambda}$ and $f(y) = (1/\lambda) e^{-y/\lambda}$. Let $U=X+Y$ and $V= X/Y$.

a.Find the joint density of $U$ and $V$.

b. Are $U$ and $V$ independent? Why?

c. Find the marginal density of $U$ and $V$.

d. Find $E[U|V=1]$

Attempt:

a. $X = U-Y$ and $Y = VX$. I cannot get $X$ or $Y$ in terms of only $U$ and $V$.

b. I know I have to find the joint distribution and see if it can be decomposed in a function of $U$ and a function of $V$.

c. For the $f(U)$, integrate with respect to $dv$ and vice-versa.

d. If they are independent, this is just equal to $E[U]$.

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My impression is that you could read with some profit this answer. –  Did Feb 28 '12 at 21:25

1 Answer 1

up vote 3 down vote accepted

I'm not sure why you cannot inver the map. Simple algebra shows that $Y=\frac{U}{V+1}$ and $X=YV=V\frac{U}{V+1}$. You should be able to employ the method of jacobian using these maps.

The rest follows by direction computation.

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