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Math Help - Multivariate Distribution function

  1. #1
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    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!!
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  2. #2
    MHF Contributor matheagle's Avatar
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    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
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  3. #3
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    Thanks matheagle! Is that the answer? Cause i thought it would involve a lot more equations to do! By the way when u mean W = Y-X It means that the value W represents the variables Y - X right? Thank you.
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