Is a moment-generating function a Fourier transform of a probability density function?
In other words, is a moment generating function just the spectral resolution of a probability density distribution of a random variable, i.e. an equivalent way to characterize a function in terms of it's amplitude, phase and frequency instead of in terms of a parameter?
If so, can we give a physical interpretation to this beast?
I ask because in statistical physics a cumulant generating function, the logarithm of a moment generating function, is an additive quantity that characterizes a physical system. If you think of energy as a random variable, then it's cumulant generating function has a very intuitive interpretation as the spread of energy throughout a system. Is there a similar intuitive interpretation for the moment generating function?
I understand the mathematical utility of it, but it's not just a trick concept, surely there's meaning behind it conceptually?
Best Answer
The MGF is
$M_{X}(t)=E\left[ e^{tX} \right]$
for real values of $t$ where the expectation exists. In terms of a probability density function $f(x)$,
$M_{X}(t)=\int_{-\infty}^{\infty} e^{tx}f(x) dx.$
This is not a Fourier transform (which would have $e^{itx}$ rather than $e^{tx}$.
The moment generating function is almost a two-sided Laplace transform, but the two-sided Laplace transform has $e^{-tx}$ rather than $e^{tx}$.