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Thread: Bivariate distribution

  1. #1
    Member roshanhero's Avatar
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    Bivariate distribution

    For two random variables X and Y,
    Prove that
    $\displaystyle 1.E(X)=E[E(X/Y)]
    2.V(X)=E(V(X/Y)]+V[E(X/Y)]$
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  2. #2
    Grand Panjandrum
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    Quote Originally Posted by roshanhero View Post
    For two random variables X and Y,
    Prove that
    $\displaystyle 1.E(X)=E[E(X/Y)]
    2.V(X)=E(V(X/Y)]+V[E(X/Y)]$

    This notation is unsatisfactory. You are using / rather than | to denote a conditional "thing".

    You have a joint distribution $\displaystyle p(x,y)$ for you RV's, then:

    $\displaystyle E(X)=\int \int x \;p(x,y)\; dy dx$

    Also

    $\displaystyle E(X|y)=\int x \; p(x|y) \;dx$

    and $\displaystyle p(x|y)=\frac{p(x,y)}{p(y)}$

    so:

    $\displaystyle E(E(X|Y))=\int E(X|y) p(y) \; dy=\int \int x\; p(x,y)\;dx\;dy$

    Now change the order of integration

    CB
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