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Math Help - Conditional covariance

  1. #1
    Senior Member Sambit's Avatar
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    Question Conditional covariance

    X_i\sim Ber(\frac{1}{4}) and Y_i\sim Ber(\frac{3}{4}). Find Cov(\sum_{i=1}^{N} X_i,\sum_{i=1}^{N} Y_i) where P(N=k)=\frac{1}{2^k} for k=1,2,3,....

    I thought using conditional expectation might be a good idea but got stuck.

    Thanks in advance.
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  2. #2
    Moo
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    Hello,

    And what's the covariance between the Xi and Yi ?

    In order not to be stuck, you may change the indices :

    cov\left(\sum_{i=1}^N X_i,\sum_{j=1}^N Y_j\right)=E_N\left[\cov\left(\sum_{i=1}^N X_i,\sum_{j=1}^N Y_j \mid N\right)\right]
    =E\left[\sum_{i=1}^N\sum_{j=1}^N cov(X_i,Y_j)\right]

    For example, if cov(X_i,Y_j)=a \quad \forall i,j, then this will be equal to a \cdot E[N^2]
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  3. #3
    Senior Member Sambit's Avatar
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    Yes; and here comes my problem: how will I calculate E(XY)? It gets complicated.
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  4. #4
    Moo
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    Quote Originally Posted by Sambit View Post
    Yes; and here comes my problem: how will I calculate E(XY)? It gets complicated.
    Because you're missing information !!!!
    We don't even care about the distributions of X and Y, we care about their joint distribution (their covariance)
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  5. #5
    Senior Member Sambit's Avatar
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    (X,Y) jointly follows a multinomial distribution with parameters N,\frac{1}{4},\frac{3}{4}. So the covariance is -N\frac{1}{4}\frac{3}{4}
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