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Thread: calculating $E(Y)$

  1. #1
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    calculating $E(Y)$

    suppose X is random variable non negative with Distribution function F_{X}(x) and Y is random variable with Distribution function G_{Y}(t)=1-E(e^{-tX}) ,0\leq t . how can i calculate E(Y)
    Last edited by Dason; 05-31-2016 at 01:33 PM.

  2. #2
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    Re: calculating $E(Y)$

    First you can easily check that $Y$ is non-negative almost surely by checking its CDF G_Y. Then the expectation of such a random variable can be expressed as

    E[Y] = \int_0^{\infty} [1 - G_Y(y)]dy

    Also, the expectation of h(X) can be expressed as

    E[h(X)] = \int_0^{\infty} h(x)dF_X(x)

    With these facts I think you can fill all the details.

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