Need help with a joint density function

**tralli** · February 18th 2010, 07:55 AM

First let me apologise for my poor english, it's not my first language.

Here is the joint density function:

$ f(y,x)=\left\{\begin{array}{cc}e^{-y},&\mbox{ if } 0 \leq x \leq y \leq 0,\\0, & \mbox{ elsewhere. } \end{array}\right. $

I'm trying to find $E(Y-X)$ and subsequenty $V(Y-X)$ .

I started by finding the marginal density functions:

$f_1(Y)= \int_{0}^{y} e^{-y} dx = xe^{-y}\Big|^{y}_{0}=ye^{-y} $

and

$f_2(X)= \int_{x}^{\infty} e^{-y} dy = -e^{-y}\Big|^{\infty}_{x} =e^{-x}$

Integration isn't really my strong suit, but I think I'm on the right path here. Next, finding $E(Y)$ and $E(X)$

$E(Y)= \int_{x}^{\infty} y^2e^{-y}dy= -\frac{y^3}{3}e^{-y}\Big|^{\infty}_{x}=\frac{x^3}{3}$

$E(X)= \int_{0}^{y} xe^{-x}dx=-\frac{x^2}{2}e^{-x}\Big|^{y}_{0}=-\frac{y^2}{2}e^{-y}$

This seems really counter-intuitive, but I'm not sure what I'm doing wrong (or right for that matter). I'd really appreciate any help. Thanks in advance.

**statmajor** · February 18th 2010, 11:36 AM

Haven't done this in a while, so I just looked on wikipedia. This is what it said (for expected value):

Expected value - Wikipedia, the free encyclopedia

result is valid even if X is not statistically independent of Y.

So in your case E(Y - X) = E(Y) - E(X).

To find your V(Y-X), you'll need to use the following property:

Variance - Wikipedia, the free encyclopedia

since in your case X and Y are not independent, their covariance is not 0.

**tralli** · February 18th 2010, 01:17 PM

Thank you.

Could you maybe look over my integrations for E(Y) and E(X). I'm not really confident I've done them correctly and would appreciate any help

**statmajor** · February 18th 2010, 01:23 PM

In your calculations of E(Y) and E(X), did you use integration by parts? If you didn't, then you made a mistake.

**tralli** · February 19th 2010, 05:24 AM

That's the mistake I made. Now it makes sense. Thanks again for your help.

**statmajor** · February 19th 2010, 06:58 AM

No problem.

Integration by parts - Wikipedia, the free encyclopedia

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