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Math Help - double expectation proof

  1. #1
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    double expectation proof

    show that (assuming all the expectatons and variances exist and are finite)

    E(Y)= E(E[Y|X])

    and

    Var (Y) = E(Var[Y|X]) + Var (E[Y|X])
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  2. #2
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    Quote Originally Posted by weakmath View Post
    show that (assuming all the expectatons and variances exist and are finite)

    E(Y)= E(E[Y|X])

    and

    Var (Y) = E(Var[Y|X]) + Var (E[Y|X])
    Let X and Y have have joint density function f(x, y) and marginal density functions f_X (x) and f_Y (y) respectively.

    Then:

    E(Y) = \int_{-\infty}^{+\infty} y \, f_Y (y) \, dy = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} y \, f(x, y) \, dx \, dy


    = \int_{-\infty}^{+\infty} \int_{-\infty}^{+\infty} y \, f(y | x) \, f_X (x) \, dy \, dx


    = \int_{-\infty}^{+\infty} \left[ \int_{-\infty}^{+\infty} y \, f(y | x) \, dy \right] f_X (x) \, dx


    = \int_{-\infty}^{+\infty} E(Y | X = x)\, f_X (x) \, dx


    = E[E(Y | X)].



    Var(Y) is left for you to try doing again.
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