# negative variance... (getting correlation from joint density)

• Jan 24th 2011, 06:45 PM
Volga
negative variance... (getting correlation from joint density)
Question. Consider random variables X and Y with joint density

$f(x)=\left\{\begin{array}{cc}8xy,&\mbox{ if }
0

Evaluate Corr (X,Y).

To use Corr (X,Y) formula, I will need to know Cov(X,Y)=E(XY)-E(X)E(Y) and Var(X) and Var(Y). This is a simple question but I keep getting negative variance for X and I re-checked several times. Help!!

$f_X(x)=\int^{\infty}_{-\infty}f_{X,Y}(x,y)dy=\int^1_x8xydy=8x[\frac{y^2}{2}]^1_x=8x(\frac{1}{2}-\frac{x^2}{2})=4x(x-x^2)=4x-x^3$ (on the interval 0<x<1, and 0 otherwise)

$E(X)=\int^{\infty}_{-\infty}xf_{X}(x)dx=\int_0^1x(4x-x^3)dx=\int_0^1(4x^2-x^4)dx=$
$=[\frac{4x^3}{4}-\frac{x^5}{5}]_0^1=4/3-1/5=17/15$

$E(X^2)=\int^{\infty}_{-\infty}x^2f_{X}(x)dx=\int_0^1x^2(4x-x^3)dx=$
$=\int_0^1(4x^3-x^5)dx=$
$=[4x^4/4-x^6/6]_0^1=1-1/6=5/6$

Then $Var(X)=E(X^2)-E(X)^2=5/6-(17/15)^2=-203/450 <0$ (Headbang)

(For Y, $f_Y(y)=4y^3$ and Var(Y)=2/75, so I should be able to proceed to find Cov and Corr if I have a positive result for Var(X)).

thanks!
• Jan 24th 2011, 07:06 PM
theodds
You fudged up calculating $f_X (x)$. It should be $4(x - x^3) \mbox{I}_{x \in [0, 1]}$.

Also, IMO, calculating the marginal of X is just asking for a stupid mistake. Calculate all the expectations from the joint, and integrate over x first so that you don't have the additive terms. It doesn't matter for the end result, but its easier and you can do it in your head.
• Jan 24th 2011, 08:41 PM
matheagle
Looks like another Walpole problem
I wouldn't get the marginals.
You should integrate all expectations using...

$\int_0^1\int_0^y ....dxdy$

or you can use

$\int_0^1\int_x^1 ....dydx$

But I prefer 0 as the lower bounds.
• Jan 24th 2011, 08:59 PM
Volga
Or dear, I must be the worst mathematician on this forum )))

Thanks for the tip, this makes sense, dealing with one integrand instead of three different ones.
• Jan 24th 2011, 09:53 PM
matheagle
Well I teach out of Walpole and I also teach multivariate calc.
IF you decide to get the marginals, then you are forcing yourself to integrate in a particular order.
What is smarter, is to figure out which was is best, dxdy or dydx, and then integrate all of these
expectations in that order.
• Jan 24th 2011, 10:27 PM
Volga
Yes, I see your point. I did study multivariable calc a couple of months ago.

Here, I followed the approach given in my Study Guide (from London School of Economics, Distribution Theory by J. Penzer) - I self-study, ie no access to lectures or tutorials. Now I consulted the texbook (I use Hogg and Tanis, and also Grimmet and Stirzaker) and they have the same alternative too.

I find this forum helps enormously in my situation, thank you all!