Laplace transform of two random variables

**tttcomrader** · June 5th 2012, 02:31 PM

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?

Thanks!

**Moo** · June 16th 2012, 08:17 AM

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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