Moment Generating Function

Definition

The MGF of RV $X$ is $m(t)=E(e^{tX})={\int }_{-\infty }^{\infty }{e}^{tx}{f}_X(x)dx$

• Well Defined when $m(t)$ is finite $\forall t<ϵ,ϵ>0$

Properties

• Linearity to multiply: $Y=a_1{X}_1+\cdots +a_n{X}_n⟹m_Y(t)=m_{X_1}(a_{1}t)\times \cdots \times m_{X_n}(a_{n}t)={\Pi }_{i=1}^n{m}_{X_i}(a_{i}t)$
• IID $X_1,X_2,\dots ,X_n,Y=X_1+\cdots +X_n⟹m_Y(t)=(m_X(t){)}^n$

Calculating Moments

$E(X^k)=m^k(0)=\frac{d^k}{d{t}^k}m(t){|}_{t=0}$