4.2. MOMENT GENERATING FUNCTIONS 131
for all valuestfor which the integral converges.
Example.Thedegenerate probability distributionhas all the probability
concentrated at a single point. That is, ifXis a degenerate random variable
with the degenerate probability distribution, thenX=μwith probability 1
andXis any other value with probability 0. That is, the degenerate random
variable is a discrete random variable exhibiting certainty of outcome. The
moment generating function of the degenerate random variable is particularly
simple: ∑
xi=μ
exit=eμt.
If the moments of orderkexist for 0≤k≤k 0 , then the moment gen-
erating function is continuously differentiable up to orderk 0 att= 0. The
moments ofXcan be generated fromφX(t) by repeated differentiation:
φ′X=
d
dt
E
[
etX
]
=
d
dt
∫
x
etxfX(x)dx
=
∫
x
d
dt
etxfX(x)dx
=
∫
x
xetxfX(x)dx
=E
[
XetX
]
.
Then
φ′X(0) =E[X].
Likewise
φ′′X(t) =
d
dt
φ′X(t)
=
d
dt
∫
x
xetxfX(x)dx
=
∫
x
x
d
dt
etxfX(x)dx
=
∫
x
x^2 etxfX(x)dx
=E
[
X^2 etX