Editing Moment-generating function (section)

==Definition==
Let <math> X </math> be a [[random variable]] with [[Cumulative distribution function|CDF]] <math>F_X</math>. The moment generating function (mgf) of <math>X</math> (or <math>F_X</math>), denoted by <math>M_X(t)</math>, is

<math display="block"> M_X(t) = \operatorname E \left[e^{tX}\right] </math>

provided this [[expected value|expectation]] exists for <math>t</math> in some open [[Neighborhood (mathematics)|neighborhood]] of 0. That is, there is an <math>h > 0</math> such that for all <math>t</math> in  <math>-h < 0 < h</math>,  <math>\operatorname E \left[e^{tX}\right] </math> exists. If the expectation does not exist in an open neighborhood of 0, we say that the moment generating function does not exist.<ref>{{cite book |last1=Casella |first1=George|last2= Berger|first2= Roger L. |title=Statistical Inference |publisher=Wadsworth & Brooks/Cole|year=1990 |page=61 |isbn=0-534-11958-1 }}</ref>

In other words, the moment-generating function of {{mvar|X}} is the [[expected value|expectation]] of the random variable <math> e^{tX}</math>. More generally, when <math>\mathbf X = ( X_1, \ldots, X_n)^{\mathrm{T}}</math>, an <math>n</math>-dimensional [[random vector]], and <math>\mathbf t</math> is a fixed vector, one uses <math>\mathbf t \cdot \mathbf X = \mathbf t^\mathrm T\mathbf X</math> instead of&nbsp;{{nowrap|<math>tX</math>:}}
<math display="block"> M_{\mathbf X}(\mathbf t) := \operatorname E \left[e^{\mathbf t^\mathrm T\mathbf X}\right].</math>

<math> M_X(0) </math> always exists and is equal to&nbsp;1. However, a key problem with moment-generating functions is that moments and the moment-generating function may not exist, as the integrals need not converge absolutely. By contrast, the [[Characteristic function (probability theory)|characteristic function]] or Fourier transform always exists (because it is the integral of a bounded function on a space of finite [[measure (mathematics)|measure]]), and for some purposes may be used instead.

The moment-generating function is so named because it can be used to find the moments of the distribution.<ref>{{cite book |last=Bulmer |first=M. G. |title=Principles of Statistics |publisher=Dover |year=1979 |pages=75–79 |isbn=0-486-63760-3 }}</ref>  The series expansion of <math>e^{tX}</math> is

<math display="block">
e^{t X} = 1 + t X + \frac{t^2 X^2}{2!} + \frac{t^3 X^3}{3!} + \cdots + \frac{t^n X^n}{n!} + \cdots.
</math>

Hence,
<math display="block">\begin{align}
M_X(t) &= \operatorname E [e^{t X}] \\[1ex]
&= 1 + t \operatorname E[X] + \frac{t^2 \operatorname E[X^2]}{2!} + \frac{t^3 \operatorname E[X^3]}{3!} + \cdots + \frac{t^n\operatorname E [X^n]}{n!}+\cdots \\[1ex]
& = 1 + t m_1 + \frac{t^2 m_2}{2!} + \frac{t^3 m_3}{3!} + \cdots + \frac{t^n m_n}{n!} + \cdots,
\end{align}</math>

where <math>m_n</math> is the {{nowrap|<math>n</math>-th}} [[moment (mathematics)|moment]]. Differentiating <math>M_X(t)</math> <math>i</math> times with respect to <math>t</math> and setting <math>t = 0</math>, we obtain the <math>i</math>-th moment about the origin, <math>m_i</math>; see {{slink|#Calculations of moments}} below.

If <math>X</math> is a continuous random variable, the following relation between its moment-generating function <math>M_X(t)</math> and the [[two-sided Laplace transform]] of its probability density function <math>f_X(x)</math> holds:

<math display="block">M_X(t) = \mathcal{L}\{f_X\}(-t),</math>

since the PDF's two-sided Laplace transform is given as

<math display="block">\mathcal{L}\{f_X\}(s) = \int_{-\infty}^\infty e^{-sx} f_X(x)\, dx,</math>

and the moment-generating function's definition expands (by the [[law of the unconscious statistician]]) to
<math display="block">M_X(t) = \operatorname E \left[e^{tX}\right] = \int_{-\infty}^\infty e^{tx} f_X(x)\, dx.</math>

This is consistent with the characteristic function of <math>X</math> being a [[Wick rotation]] of <math>M_X(t)</math> when the moment generating function exists, as the characteristic function of a continuous random variable <math>X</math> is the [[Fourier transform]] of its probability density function <math>f_X(x)</math>, and in general when a function <math>f(x)</math> is of [[exponential order]], the Fourier transform of <math>f</math> is a Wick rotation of its two-sided Laplace transform in the region of convergence. See [[Fourier transform#Laplace transform|the relation of the Fourier and Laplace transforms]] for further information.