Multivariate Distribution function

• Mar 21st 2010, 06:20 PM
Redeemer_Pie
Multivariate Distribution function
Hello all i am having trouble breaking down this question. Anyone can help is much appreciated:

Let X = (X,Y) and is N(0, "summation sign") with "summation sign" :

| 1 p |
| p 1 |

Write an integral expression for P(X "subset sign" D) where D is a set on the plane. Evaulate it for D = {(x,y) : x <= y} by using the theorem:

If X is a multivariate Normal N (u, "summation sign") then aX is (au, a"summation sign"a"transpose sign")

Im not too sure about the theorem bit which i stated above cause there might have been a typo in the book.

Incase it is too vague there are hints given that says:

The whole idea is to Transform X and Y into standard normal variables (the formula for Z which is used to look up in the normal distribution table) so it can be used to find out the the probability of P(X,Y <= 0).

oh and u is the mean ie the "mean sign"

Thank you!!
• Mar 21st 2010, 09:57 PM
matheagle
You can express this as a double integral...

$P(Y>X)= \int_{-\infty}^{\infty}\int_{-\infty}^y f(x,y)dxdy$

or let $a=(-1,1)$....

so $a(X,Y)^t=-X+Y$. Thus $P((-1,1)(X,Y)^t>0)$

is just $P(-X+Y>0)$ or $P(Y>X)$

NOW $W=Y-X$ is a normal with mean zero and ...(you get the variance, but that's not needed)

So P(W>0)=.5
• Mar 22nd 2010, 05:14 AM
Redeemer_Pie
Thanks matheagle! Is that the answer? Cause i thought it would involve a lot more equations to do! :D By the way when u mean W = Y-X It means that the value W represents the variables Y - X right? Thank you.