132 CHAPTER 4. LIMIT THEOREMS FOR STOCHASTIC PROCESSES
Then
φ′′X(0) =E[
X^2
]
.
Continuing in this way:φ(Xn)(0) =E[Xn]In words: the value of thenth derivative of the moment generating function
evaluated at 0 is the value of thenth moment ofX.
Theorem 8.IfX andY are independent random variables with moment
generating functionsφX(t)andφY(t)respectively, thenφX+Y(t), the moment
generating function ofX+Y is given byφX(t)φY(t). In words, the moment
generating function of a sum of independent random variables is the product
of the individual moment generating functions.
Proof.Using the lemma on independence above:
φX+Y(t) =E[
et(X+Y)]
=E
[
etXetY]
=E
[
etX]
E
[
etY]
=φX(t)φY(t).Theorem 9.If the moment generating function is defined in a neighborhood
oft= 0then the moment generating function uniquely determines the prob-
ability distribution. That is, there is a one-to-one correspondence between
the moment generating function and the distribution function of a random
variable, when the moment-generating function is defined and finite.
Proof.This proof is too sophisticated for the mathematical level we have
now.
The moment generating function of a normal random variable
Theorem 10.IfZ∼N(μ,σ^2 ), thenφZ(t) = exp(μt+σ^2 t^2 /2).