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Math Help - covariance and corellation

  1. #1
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    covariance and corellation

    1) Suppose that two random variables X & Y are jointly distributed according to the pdf f(x,y)=8xy , 0<=y<=x<=1. What is the correlation coefficient of X & Y?

    2) Consider the sample space S={(-2,4),(-1,1),(0,0),(1,1),(2,4)}.Where each point are assumed to be equally likely.Define the random variable X to be the first component of a sample point and Y to be the second.Then X(-2,4)=-2, Y(-2,4)=4 and so on.Is X and Y independent? What is the covariance Cov(X,Y)?
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  2. #2
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    Quote Originally Posted by thandu3 View Post
    1) Suppose that two random variables X & Y are jointly distributed according to the pdf f(x,y)=8xy , 0<=y<=x<=1. What is the correlation coefficient of X & Y?

    [snip]
    \rho = \frac{Cov(X, Y)}{\sigma_X \sigma_Y} where Cov(X, Y) = E(XY) - E(X) E(Y).

    E(XY) = \int_{x=0}^1 \int_{y=0}^{y=x} (xy) \, 8xy \, dy \, dx.

    E(X), E(Y), \sigma_X and \sigma_Y are calculated from the marginals f_X(x) = \int_{y = 0}^{y = x} 8xy \, dy and f_Y(y) = \int^{x=1}_{x=y} 8xy \, dx.
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  3. #3
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    Quote Originally Posted by thandu3 View Post
    [snip]
    2) Consider the sample space S={(-2,4),(-1,1),(0,0),(1,1),(2,4)}.Where each point are assumed to be equally likely.Define the random variable X to be the first component of a sample point and Y to be the second.Then X(-2,4)=-2, Y(-2,4)=4 and so on.Is X and Y independent? What is the covariance Cov(X,Y)?
    I suggest that you calculate the covariance first: Cov(X, Y) = E(XY) - E(X) E(Y).

    If it's not equal to zero then you know that X and Y are not independent.

    However ..... if it is equal to zero then it's still unknown whether X and Y are independent or not. In which case you should consider doing a couple of simple calculations eg. Does Pr(Y = 1) equal Pr(Y = 1 | X = 0) .... ?
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    thanks

    Thanks a lot for your kind reply.Thanks for the idea.I was working on it but an idea can make it solve quickly.
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