Moment Generating Functions
MGF Properties
The \(k\)th moment is defined as the expectation of the random variable, raised to the \(k\)th power, defined as \(E(X^k)\).
The moment generating functions is used to obtain the \(k\)th moment. The mgf is defined as
\[ m(t) = E(e^{tX}) \]
The \(k\)th moment can be obtained by taking the \(k\)th derivative of the mgf, with respect to \(t\), and setting \(t\) equal to 0:
\[ E(X^k)=\frac{d^km(t)}{dt}\Bigg|_{t=0} \]
Let \(X\) follow a distribution \(f\), with the an MGF \(M_X(t)\), the MGF of \(Y=aX+b\) is given as
\[ M_Y(t) = e^{tb}M_X(at) \]
Let \(X\) and \(Y\) have the following distributions \(F_X(x)\) and \(F_Y(y)\) and MGFs \(M_X(t)\) and \(M_Y(t)\), respectively. \(X\) and \(Y\) have the same distribution \(F_X(x)=F_Y(y)\) if and only if \(M_X(t)=M_Y(t)\).
\(X\sim\mathrm{U(a,b)}\)
\(a<x<b\)
\(f_X(x) = \frac{1}{b-a}\)
\(X\sim\mathrm{N}(\mu, \sigma^2)\)
\(-\infty < x < \infty\)
\(f_X(x) = \frac{1}{\sqrt{2\pi \sigma^2}}\exp\left\{-\frac{(x-\mu)^2}{2\sigma^2}\right\}\)
\(X\sim\mathrm{Gamma}(\alpha, \beta)\)
\(0<x<1\)
\(f_X(x)=\frac{1}{\Gamma(\alpha)\beta^\alpha}x^{\alpha-1}e^{-x/\beta}\)
\(X\sim\chi^2_k\)
\(x>0\)
\(f_X(x)=\frac{x^{k/2-1}\exp\{-x/2\}}{2^{k/2}\Gamma(k/2)}\)