# Characteristic and moment generating function of a random variable interpretation

I have been studying about moments and cumulants of a random variable. Even though the definitions of characteristic and moments generating function are very similar (only the sign in the exponential is changed) to Fourier and Laplace transforms, respectively.

$\Phi_x(\xi) = E [ e^{j \xi x(\zeta)} ] = \int_{-\infty}^{\infty} f_x(x) e^{j \xi x} dx$

$\Phi_x(s) = E [ e^{sx(\zeta)} ] = \int_{-\infty}^{\infty} f_x(x) e^{sx} dx$

The book I've been reading says the variable ΞΎ is not and should not be interpreted as frequency!

How is it supposed to be interpreted? The functions $\Phi_x(\xi)$ and $\Phi_x(s)$ possess all the properties associated with the Fourier and Laplace transforms, respectively.

Were these exponentials placed there just to be able to perform the Taylor series decomposition and then take the mth-order derivative of every term and extract the appropriate mth-order moment ?

Anyway, one could consider the "bandwidth" for $\Phi_X$ in the $\xi$-domain as how spread the pdf is around the mean. A more spread pdf would imply a narrower "bandwidth", and less spread of the pdf would imply a wider "bandwidth". For example, consider a uniformly distributed random variable: $$f_X(x) = 1/\delta~\Pi(x/\delta)$$ $$\Phi_X(\xi) = sinc(\delta~\xi)$$ As $\delta$ gets smaller, the random variable is constrained in a narrower interval, while the "bandwidth" in $\Phi_X(\xi)$ gets wider, and vice versa (measuring the "bandwidth" as the width of the main lobe, $2/\delta$, as we do for communication signals).