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Math Help - negative variance... (getting correlation from joint density)

  1. #1
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    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 }<br />
0<x<y<1\\0, & \mbox{ if } otherwise\end{array}\right

    Evaluate Corr (X,Y).

    Answer.

    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

    (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!
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  2. #2
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    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.
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  3. #3
    MHF Contributor matheagle's Avatar
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    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.
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  4. #4
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    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.
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  5. #5
    MHF Contributor matheagle's Avatar
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    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.
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  6. #6
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    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!
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