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Math Help - Finding Covariance of simple joint DF

  1. #1
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    Finding Covariance of simple joint DF

    A joint density function is given by:
    f(x, y) = kx 0 < x, y < 1
    0 otherwise
    Find Cov(X, Y)
    ******
    Those bounds are giving me trouble. At first I read it as 0 < x < y < 1. But I get 2 different values for k depending if I use the direction of f(x) or f(y)...

    I found:
    f(x) = k - kx^2 which lead to k = 3/2
    f(y) = (ky^2)/2 which lead to k = 6
    I'm not sure if it's possible to have 2 different values of k like that ?
    E(xy) led to another problem, which k to chose ? So I must have done something wrong with those bounds...

    Thanks !!!
    The Answer is 0
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  2. #2
    MHF Contributor harish21's Avatar
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    Re: Finding Covariance of simple joint DF

    Quote Originally Posted by TheCountador View Post
    A joint density function is given by:
    f(x, y) = kx 0 < x, y < 1
    0 otherwise
    Find Cov(X, Y)
    ******
    Those bounds are giving me trouble. At first I read it as 0 < x < y < 1. But I get 2 different values for k depending if I use the direction of f(x) or f(y)...

    I found:
    f(x) = k - kx^2 which lead to k = 3/2
    f(y) = (ky^2)/2 which lead to k = 6
    I'm not sure if it's possible to have 2 different values of k like that ?
    E(xy) led to another problem, which k to chose ? So I must have done something wrong with those bounds...

    Thanks !!!
    The Answer is 0
    No you cant have 2 different values of k.

    you are given that f(x,y) = k\cdot x\;\;0<x,y<1

    that means 0<x<1 \;and\; 0<y<1; x and y range between 0 and 1, but you are not given that x<y.

    so f(x) = \int_y f(x,y) dy= \int_0^1 kx \; dy= kx

    f(y)=\int_x f(x,y) dx = \int_0^1 kx dx = \frac{k}{2}

    now you can find the value of k by using the fact that \int_0^1 f(x)\;dx = 1 and \int_0^1 f(y) dy =1

    you can then plug the value of k in f(x,y) and integrate to see that it integrates to 1.
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  3. #3
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    Re: Finding Covariance of simple joint DF

    I've got it thank you. That comma in the range threw me off
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  4. #4
    MHF Contributor harish21's Avatar
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    Re: Finding Covariance of simple joint DF

    but k cannot be 0.

    edit: Ignore this post. looks like your covariance is 0. I thought you were saying k=0
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