Editing Moment-generating function (section)

==Relation to other functions==
Related to the moment-generating function are a number of other [[integral transform|transforms]] that are common in probability theory:

;[[Characteristic function (probability theory)|Characteristic function]]: The [[characteristic function (probability theory)|characteristic function]] <math>\varphi_X(t)</math> is related to the moment-generating function via <math>\varphi_X(t) = M_{iX}(t) = M_X(it):</math> the characteristic function is the moment-generating function of ''iX'' or the moment generating function of ''X'' evaluated on the imaginary axis.  This function can also be viewed as the [[Fourier transform]] of the [[probability density function]], which can therefore be deduced from it by inverse Fourier transform.
;[[Cumulant-generating function]]: The [[cumulant-generating function]] is defined as the logarithm of the moment-generating function; some instead define the cumulant-generating function as the logarithm of the [[Characteristic function (probability theory)|characteristic function]], while others call this latter the ''second'' cumulant-generating function.
;[[Probability-generating function]]: The [[probability-generating function]] is defined as <math>G(z) = \operatorname{E}\left[z^X\right].</math> This immediately implies that <math>G(e^t) = \operatorname{E}\left[e^{tX}\right] = M_X(t).</math>