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Math Help - Laplace transform of two random variables

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    Laplace transform of two random variables

    Let X and Y be random variables with distributions F and G; and write the laplace transforms of them as  \hat F , \hat G

    Show that  E[e^{- \lambda xy }] = \int ^ \infty _0 \hat F (\lambda y)dG(y) = \int ^ \infty _0 \hat G( \lambda y)dF(y)

    I'm just confused about the expected value of two different variables, does that means I should use dF or dG as the probability density function for it?

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    Re: Laplace transform of two random variables

    Hello,

    It should be XY in the expectation, not xy. Since the random variables are named X and Y.

    When you have the expectation containing 2 or more random variables, you must take the joint pdf of the variables (joint pdf of (X,Y) here).

    For your question, consider the conditional distributions.
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