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Moment generating functions provide an alternative specification for a probability distribution function (pdf), often making it very convenient to calculate expectations, variances, etc. of said pdf.

A moment generating function is defined as:

$M_X(t) = \mathbb{E}(e^{tX}), t\in\mathbb{R}$

A caveat to moment generating functions is that their integrals might not always exist. The more general characteristic function gets around this, and they are defined as:

$C_X(t) = \mathbb{E}(e^{itX}), t\in\mathbb{R}$

Given their tremendous capacity for simplifying the calculation of the moments of a probability distrubtion, moment-generating and characteristic functions seem almost magical -- a wonderful rabbit pulled, from thin air, out of a magician's hat.

Few things are truly that way in the history of mathematics, so I am curious to know how this particular rabbit was born: how did human beings zero in on them?

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  • $\begingroup$ I do not have enough reputation to comment, but the book generatingfunctionology by Herbert Wilf is well written, fun to read, and connects the gap "between discrete mathematics and ... continuous analysis." While the author does not attempt to give a comprehensive overview of the subject, I found value in reading about Wilf's approach to the seemingly ever-applicable world of generating functions. $\endgroup$
    – polykleitos
    Sep 30 at 16:02
  • $\begingroup$ For such historical information, perhaps try W. Feller, An Introduction to Probability Theory and Its Applications, Vol. 1 $\endgroup$
    – Gerald Edgar
    Oct 1 at 1:23

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The general idea of generating function has much wider scope than its applications to probability. The proper setting is ``harmonic analysis'' which is one of the central and most developed parts of mathematics. The birth of the idea can be traced back to Abraham de Moivre (1667-1754), and his book Doctrine of Chances. Later the same idea was developed and applied in number theory (Euler), and most importantly in mathematical physics (Fourier). (Characteristic function is a special case of Fourier transform). Laplace transform (the moment generating function) belongs to the same circle of ideas, and its original use was also in probability.

All this powerful set of ideas (Generating function, Fourier, Laplace and other similar transforms) is called (linear) Harmonic analysis and it is one of the most powerful methods of mathematics, with applications practically everywhere (in almost all areas of mathematics and sciences). If you add a generalization of it, called non-linear harmonic analysis, you obtain most of the existing mathematics:-)

There is an excellent historical survey:

Mackey, George W. Harmonic analysis as the exploitation of symmetry—a historical survey. Bull. Amer. Math. Soc. 3 (1980), no. 1, part 1, 543–698

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    $\begingroup$ Oh my god -- what a wonderful article (and a nice answer, too :)! $\endgroup$
    – bzm3r
    Feb 14, 2016 at 21:01
  • $\begingroup$ I have a question from the paper, which I posted on M.SE -- hope you can help! math.stackexchange.com/questions/1657651/… $\endgroup$
    – bzm3r
    Feb 16, 2016 at 2:41

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