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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 $E[U]$.

Michael Hardy
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lord12
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2 Answers2

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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.

user25720
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For b: there are two ways to solve this: You have to fatorate the joint function of u and v and show that it is the product of the u density and the v. Other way is to integrate the joint function in order to find both marginals and then multiply them, if the result equals the joint function then they are independent

For c: Integrate the joint function in the whole domain of some variable to find the others marginal

For d: calculate normally using the conditional density!

Beware the integration domain! But in this case , as the variables are exponential this won't be such a problem i hope ! Wish i've helped

Ariel Serranoni
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