I would gladly appreciate any and all help with this joint density problem a practice problem for an exam. Please excuse my lack of use of the proper symbols, I don't know how to express them online unfortunately :
Joint Density of two random variables, U and V is:
f(u,v)= 1/u for 0<v<u, and 0<u<1
1. Find marginal densities of U and V.
For marginal density of u, I did the integral of 1/u with respect to v from 0 to u. I got the integral to be v/u. And then when you plug in 0 and u, I got 1. This doesn't look right to me at all.
For the marginal density of v, I did the the integral of 1/u with respect to u and I got ln(u). I then plugged in 0 and 1 and I got 0. For the ln u, when I plugged in the 0, I just did the limit as u approaches 0 and that is zero. So it should be 0-0, which is zero.
Does this even look remotely correct? I don't believe it is.
2. Find E(u) and E(v)
For E(u), I just did the double integral (from 0 to 1 and 0 to u) of 1, since u*(1/u) equals 1. I got 1/2.
For E(v), I did the double integral (from 0 to 1 and 0 to u) of v/u and I got 1/8.
Again, these answers don't look right. Any and all help is GREATLY appreciated.