A Few Conceptual Questions About Laplace Transforms and Moment Generating Functions

Question

I have a few quick questions designed to understand Laplace Transforms and Moment Generating Functions better.

1. Is the formulaic way to go from a Moment Generating Function to a Probability Density Function or a Probability Mass Function a line integral in the complex plane, analogous to the Inverse Laplace Transform?

2. Does taking the $n$-th derivative of a Laplace Transform (such as in the context of Differential Equations) with respect to the frequency domain variable $s$ and evaluating the result at $s = 0$ yield anything interesting, analogous to how the Moment Generating Function yields raw moments?

3. Is there a geometric intuition to explain why the way to invert an integral transform is another integral transform, rather than taking some sort of derivative (as the fundamental theorem of calculus would predict)?

Answer 1

1. The MGF is $\mathbb{E}(\exp(tX))$ and, for a discrete random variable, the probability generating function (whose coefficients are the PMF) is $\mathbb{E}(t^X)$, so passing from one to the other amounts at least formally to a substitution $t \mapsto \log t$. If $X$ is a continuous random variable with a PDF $f$ then its MGF is precisely the (bilateral) Laplace transform of $f$ (up to perhaps a sign) so you get $f$ back via precisely an inverse Laplace transform.

2. It's the same; again at least formally you get the "moments" of the original function $f$ back, that is, the integrals $\int t^n f(t) \, dt$. This follows from observing that the Laplace transform intertwines multiplication by $t$ and differentiation with respect to $s$ (again up to a sign).

3. In the fundamental theorem of calculus the new function you get is a function of the upper bound in the integral. In an integral transform the new function you get is a function of a parameter you've inserted into the integral. So the two situations are not as similar as they seem. That's not really a complete answer though. To build intuition you may want to spend some time learning about the discrete Fourier transform and trying to get some sense for how it produces Fourier series and the Fourier transform, at least formally, under appropriate limits.