Editing Moment-generating function (section)

==Calculation==
The moment-generating function is the expectation of a function of the random variable, it can be written as:

* For a discrete [[probability mass function]], <math>M_X(t)=\sum_{i=0}^\infty e^{tx_i}\, p_i</math>
* For a continuous [[probability density function]], <math> M_X(t)  = \int_{-\infty}^\infty e^{tx} f(x)\,dx </math>
* In the general case: <math>M_X(t) = \int_{-\infty}^\infty e^{tx}\,dF(x)</math>, using the [[Riemann&ndash;Stieltjes integral]], and  where <math>F</math> is the [[cumulative distribution function]]. This is simply the [[Laplace-Stieltjes transform]] of <math>F</math>, but with the sign of the argument reversed.

Note that for the case where <math>X</math> has a continuous [[probability density function]] <math>f(x)</math>,  <math>M_X(-t)</math> is the [[two-sided Laplace transform]] of <math>f(x)</math>.

<math display="block">\begin{align}
M_X(t) & = \int_{-\infty}^\infty e^{tx} f(x)\,dx \\[1ex]
& = \int_{-\infty}^\infty \left( 1+ tx + \frac{t^2 x^2}{2!} + \cdots + \frac{t^n x^n}{n!} + \cdots\right) f(x)\,dx \\[1ex]
& = 1 + tm_1 + \frac{t^2 m_2}{2!} + \cdots + \frac{t^n m_n}{n!} +\cdots,
\end{align}</math>

where <math>m_n</math> is the <math>n</math>th [[moment (mathematics)|moment]].

===Linear transformations of random variables ===
If random variable <math>X</math> has moment generating function <math>M_X(t)</math>, then <math>\alpha X + \beta</math> has moment generating function <math>M_{\alpha X + \beta}(t) = e^{\beta t}M_X(\alpha t)</math>

<math display="block">
M_{\alpha X + \beta}(t) = \operatorname{E}\left[e^{(\alpha X + \beta) t}\right] = e^{\beta t} \operatorname{E}\left[e^{\alpha Xt}\right] = e^{\beta t} M_X(\alpha t)
</math>

===Linear combination of independent random variables===
If <math display="inline">S_n = \sum_{i=1}^n a_i X_i</math>, where the {{math|''X''<sub>''i''</sub>}} are independent random variables and the {{math|''a''<sub>''i''</sub>}} are constants, then the probability density function for {{math|''S''<sub>''n''</sub>}} is the [[convolution]] of the probability density functions of each of the {{math|''X''<sub>''i''</sub>}}, and the moment-generating function for {{math|''S''<sub>''n''</sub>}} is given by

<math display="block">
M_{S_n}(t) = M_{X_1}(a_1t) M_{X_2}(a_2t) \cdots M_{X_n}(a_nt) \, .
</math>
<!----------
Below was lifted from [[generating function]] ... there should be an 
analog for the moment-generating functionbuted with common probability-generating function ''G''<sub>X</sub>, then
<math display="block">G_{S_N}(z) = G_N(G_X(z)).</math>
-------->

===Vector-valued random variables===
For [[random vector|vector-valued random variables]] <math>\mathbf X</math> with [[real number|real]] components, the moment-generating function is given by

<math display="block"> M_X(\mathbf t) = \operatorname{E}\left[e^{\langle \mathbf t, \mathbf X \rangle}\right] </math>

where <math>\mathbf t</math> is a vector and <math>\langle \cdot, \cdot \rangle</math> is the [[dot product]].