Moment generating functions
The moment generating function (mgf) of a random variable is defined as \[M_X(t)= E[\exp{(t )}]~~~~ \mbox{if this exists.}\] $$E\left[X^k\right]$$ can be evaluated as the:
(i) coefficient of $$\displaystyle\frac{t^r}{r !}$$ in the power expansion of $$M_X(t)$$;
(ii) $$r$$-th derivative of $$\displaystyle M_X(t)$$ evaluated at $$t = 0.$$