Slides Chapter 1. Measure Theory and Probability

1.9. THE CHARACTERISTIC FUNCTION
(b) The moments α n are the same ∀a ∈ [0,1].
(c) The series ∑ ∞
n=0 α n (it)n
n!
diverges for all t ≠ 0.
Definition 1.53 (Moment generating function)
We define the moment generating function (m.g.f.) of a r.v. X as
M(t) = E [ e tX] , t ∈ (−r,r) ⊂ IR, r > 0,
assuming that there exists r > 0 such that the integral exists for
all t ∈ (−r,r). If such r > 0 does not exist, then we say that the
m.g.f. of X does not exist.
Remark 1.18 Remember that the c.f. always exists unlike the
m.g.f.
Proposition 1.22 (Properties of the m.g.f.)
If there exists r > 0 such that the series
∞∑ (tx) n
n=0
is uniformly convergent in (−r,r), where r is called the radius of
convergence of the series, then it holds that
(a) The n-th moment α n exists and if finite, ∀n ∈ IN;
(b) The n-th derivative of M(t), evaluated at t = 0, exists and it
satisfies M n) (0) = α n , ∀n ∈ IN;
(c) M(t) can be expressed as
∞∑
M(t) =
n=0
n!
α n
n! tn , t ∈ (−r,r).
Remark 1.19 UndertheassumptionofProposition1.22,themoments
{α n } ∞ n=0 determine the d.f. F.
ISABEL MOLINA 53