moment generating function

E[X^k] is k-th order moment. 

E[e^(tX)] is called moment generating function.

(the literal meaning is expected value of e^(tX))

e^(tX) contains all X^k. it is obvious because, by applying Taylor series for e^(tX),

e^(tX) = 1+t X+t^2 X^2/2+t^3 X^3/6+t^4 X^4/24+t^5 X^5/120+O(X^6).