Moment Generation Function

The Moment Generating function is a Mathematical tool that can establish a relation between the Moment and the PDF of a Random variable. 

Mathematically

$$\begin{array}{rl}{M}_X(t)& =\mathbb{E}[e^{tX}]\\ & =\mathbb{E}\left[1+t{X}^1+\frac{t^2}{2!}{X}^2+\frac{t^3}{3!}{X}^3...\right]\\ & =1+t{\mathbb{E}}^1[X]+\frac{t^2}{2!}{\mathbb{E}}^2[X]+\frac{t^3}{3!}{\mathbb{E}}^3[X]\end{array}$$ $$\boxed{M_X(t)=1+t{\mathbb{M}}^1[X]+\frac{t^2}{2!}{\mathbb{M}}^2[X]+\frac{t^3}{3!}{\mathbb{M}}^3[X]}$$

Note: If all the moments of a Random variable is known, then the PDF of a Random variable can be calculated using the moment generating function. However the relation is complicated, hence omitted here.