Moment Generating Function

Statements

Definition. Let $X$ be a random variable with distribution $P_X$, the expected value

$$E(X^n)=\{\begin{array}{llc}\sum _{i\in [1,N]}{x}_i^n{P}_X(x_i),& \text{if}& P_X\text{is discrete}\\ \int x^n{P}_X(x)dx,& \text{if}& P_X\text{is continuous}\end{array}$$

is said to the $n-$th raw moment. Let $\mu =E(X)$, the $n-$th central moment is given by

$$E((X-E(X){)}^n)=\{\begin{array}{llc}\sum _{i\in [1,N]}(x_i-\mu {)}^n{P}_X(x_i),& \text{if}& P_X\text{is discrete}\\ \int (x-\mu {)}^n{P}_X(x)dx,& \text{if}& P_X\text{is continuous}.\end{array}$$

Remarks

Remark. The 1st raw moment reduces to the expected value.

Remark. The 2nd central moment reduces to the variance.