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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 ξ 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 ?

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As you well said, these functions are useful just for their mathematical properties that allow to compute the m-th order moment of a random variable.

Additionally, they happen to match Fourier transform theorems. However, not in the sense of a time-domain signal, since the variable in the probability density function (pdf) is the random variable itself, and not time. A frequency interpretation only makes sense in time-dependent functions. Actually the definition itself of frequency is how often something happens.

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).

Just by observing the characteristic function one may have an idea of how spread is the pdf. By comparing two characteristic functions with different "bandwidths" one may determine which pdf is spread the most.

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Hi: Since the characteristic function of any random variable uniquely defines the distribution associated with the random variable ( that's a theorem but I forget the name of it ), the cf can be used to identify the distribution of some function of a random variable. ( e.g sum of n exponentials is chisquared(n) ). As you said, the moment generating function is used to derive the moments of an R.V. So, yes, nothing to do with frequency domain.

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