Moment Generating Function

Definition

Univariate

Let $X$ be a Random Variable $M_X(t)=E(e^{tX})=\sum _{i=0}^{\infty }\frac{E(X^i)}{i!}{t}^i$

Then, $M_X(t)$ is called the moment generating function of the $X$

Calculations of Moments

${\mu }_n^{\prime }=E(X^n)=M_X^{(n)}(0)$

Moment can be calculated using the mgf

Uniqueness of MGF

Let $X,Y$ be random variables with mgf $M_X(t),M_Y(t)$, respectively Then, $(\forall z\in \mathbb{R},F_X(z)=F_Y(z))\iff (\forall t\in (-h,h),M_X(t)=M_Y(t))$ where $h>0$

Let $X$ be a Random Variable If $\exists M_X(t),\forall t\text{s.t.}||t||\le t_0$, where $t_0\in {\mathbb{R}}^{+}$, then $M_X(t)$ determine the distribution uniquely.

Multivariate

Let $\mathbf{X}=(X_1,X_2,\dots ,X_n)$ be a Random Vector $M_{\mathbf{X}}(\mathbf{t})=E(e^{{\mathbf{t}}^{\prime }\mathbf{X}})$ where $\mathbf{t}=(t_1,\dots ,t_n)$ is a constant vector.

Calculations of Multivariate Moments

$\frac{{\partial }^{k+m}M(t_1,t_2)}{\partial t_1^k\partial t_2^m}{|}_{t_1=t_2=0}={\int }_{-\infty }^{\infty }{\int }_{-\infty }^{\infty }{x}^k{y}^{m}f(x,y)dxdy=E(X^k{Y}^m)$

Moment can be calculated using the mgf

Marginal MGF

$M_{\mathbf{X}}(t_1,0,\dots ,0)=M_{X_1}(t_1)$

Facts

The mgf may not exist.

Let $\mathbf{X}=(X_1,X_2,\dots ,X_n)$ be a Random Vector. If $M_{\mathbf{X}}(\mathbf{t})$ exists and can be expressed as $M_{\mathbf{X}}(\mathbf{t})=M_{\mathbf{X}}(t_1,\dots ,t_r,0,\dots ,0)M_{\mathbf{X}}(0,\dots ,0,t_{r+1},\dots ,t_n)$ then ${\mathbf{X}}_1=(X_1,\dots ,X_r{)}^{⊺}$ and ${\mathbf{X}}_2=(X_{r+1},\dots ,X_n{)}^{⊺}$ are statistically independent.

Let $\mathbf{X}=(X_1,X_2,\dots ,X_n)$ be a Random Vector, ${\mathbf{X}}_1=(X_1,\dots ,X_r{)}^{⊺}$, and ${\mathbf{X}}_2=(X_{r+1},\dots ,X_n{)}^{⊺}$, then ${\mathbf{X}}_1$ and ${\mathbf{X}}_2$ is independent $\iff \exists a,b\text{s.t.}{M}_{\mathbf{X}}(\mathbf{t})=a(t_1,\dots ,t_r)b(t_{r+1},\dots ,t_n)$ where $a$ and $b$ are functions.