# Joint density problem: what am I doing wrong?

• December 10th 2010, 08:40 AM
hwill205
Joint density problem: what am I doing wrong?
Hello,

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.
• December 10th 2010, 10:36 PM
CaptainBlack
Quote:

Originally Posted by hwill205
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.

That is correct

Quote:

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.
You have the wrong limits of integration, the integral should be from v=u to 1

CB
• December 11th 2010, 05:16 AM
harish21
Quote:

Originally Posted by hwill205

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.

you dont need double integration.. just integrate u times the marginal pdf of u and v times the marginal pdf of v.

$E(u) = \displaystyle \int_U u\;f(u)\;du$

and $E(v)=\displaystyle \int_V v\;f(v)\;dv$