# Laplace transform and density

It is true that the Laplace transform of a (positive) random variable characterises that random variable, just like its density?

($L_X(z) = E(exp(-Xz))$)

Up to a change of sign in the exponent, the Laplace transform is the moment generating function.

The MGF only characterizes a random variable if the MGF converges in an open interval around 0.

This isn't automatically the case, even when all moments exist (e.g. see the lognormal).

By contrast the characteristic function (which up to sign in the exponent is the same as the Fourier transform) does characterize a random variable.

• "If $L_X(\cdot) = L_Y(\cdot)$, then $X$ and $Y$ have the same distribution." This statement is wrong in general, right? Jul 9, 2014 at 10:19
• The thing is, if the integrals for $E(e^{-Xz})$ converge (so that it's meaningful to compare them for equality) then the equality of the two in an open interval around 0 would imply they do have the same distribution. Jul 9, 2014 at 10:32