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Math Help - Correlation and variance and covariance calculation CRV...?

  1. #1
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    Correlation and variance and covariance calculation CRV...?

    Hi !
    The joint pdf of X and Y is fyx = 1 for 0<x<1, x<y<x+1
    Compute the covariance and the correlation of X and Y.
    ********
    I figured out the covariance and checked that it is indeed 1/12.
    So, my E[X] = 1/2 and E[Y] = 1 are correct too.
    For E[Y] = integral(0 to 2)integral(y-1 to y) of y dxdy = 1 (this is correct, just to help with bounds)

    I can't figure out the ****correlation**** though.
    Corr(X,Y) = Cov(X,Y)/[sqrt(Var[X]Var[Y])]

    For the variances I had
    Var[X] = 1/3 - 1/2^2 = 1/12
    Var[Y] = 8/3 - 1^2 = 5/3

    My answer came to Corr(X,Y) = 0.2227 but that's wrong !!! and i can't find the mistake...

    THANKS !!!
    Answer is 0.7071
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  2. #2
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    Re: Correlation and variance and covariance calculation CRV...?

    Quote Originally Posted by TheCountador View Post
    Hi !
    The joint pdf of X and Y is fyx = 1 for 0<x<1, x<y<x+1
    Compute the covariance and the correlation of X and Y.
    ********
    I figured out the covariance and checked that it is indeed 1/12.
    So, my E[X] = 1/2 and E[Y] = 1 are correct too.
    For E[Y] = integral(0 to 2)integral(y-1 to y) of y dxdy = 1 (this is correct, just to help with bounds)

    I can't figure out the ****correlation**** though.
    Corr(X,Y) = Cov(X,Y)/[sqrt(Var[X]Var[Y])]

    For the variances I had
    Var[X] = 1/3 - 1/2^2 = 1/12
    Var[Y] = 8/3 - 1^2 = 5/3

    My answer came to Corr(X,Y) = 0.2227 but that's wrong !!! and i can't find the mistake...

    THANKS !!!
    Answer is 0.7071
    if you are getting E[X] = 1/2 and E[Y]=1, that means you have f(x)=1, and f(y)=1 (this is what you would get when you calculated your marginal pdfs from the joint pdf). Also since f(x,y)=1, that means X and Y would be independent, and covariance=correlation=0. Is your question posted correctly?
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  3. #3
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    Re: Correlation and variance and covariance calculation CRV...?

    Yes, my question is posted word for word and the covariance is definitely 1/12.
    f(x,y) = constant is independent even tho Y is bounded by functions of X ?
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  4. #4
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    Re: Correlation and variance and covariance calculation CRV...?

    Quote Originally Posted by TheCountador View Post
    Yes, my question is posted word for word and the covariance is definitely 1/12.
    f(x,y) = constant is independent even tho Y is bounded by functions of X ?
    So this is what I get E[X]=\frac{1}{2},\; E[X^2] = \frac{1}{3} and E[Y]=1\;and E[Y^2]=\frac{7}{6} which will give

    V(X)=\frac{1}{12}\;and\;V(Y)=\frac{1}{6}

    E[XY]=\frac{7}{12}

    Cov(X,Y)=\frac{7}{12}-\frac{1}{2}\cdot 1 =\frac{1}{12}

    \rho_{X,y}=\dfrac{\frac{1}{12}}{\sqrt{\frac{1}{12}  \frac{1}{6}}}=0.707

    which means you made a mistake while computing E[Y^2]
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  5. #5
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    Re: Correlation and variance and covariance calculation CRV...?

    I think my bounds are wrong (they often are!) but that's what my sketch looks like :/
    x goes from 0 to 1 and is bounded by the lines y=x and y=2x...

    E[y^2] = I(0 to 2)I(y-1 to y) y^2 dxdy = I(0 to 2) y^2 dy = 8/3 ...
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  6. #6
    MHF Contributor harish21's Avatar
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    Re: Correlation and variance and covariance calculation CRV...?

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  7. #7
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    Re: Correlation and variance and covariance calculation CRV...?

    Got it thanks again !
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