Slides Chapter 1. Measure Theory and Probability

1.9. THE CHARACTERISTIC FUNCTION
Then, if ϕ(t) is the c.f. of a r.v. X, the mean and variance are
equal to
E(X) = α 1 = ϕ′ (0)
i
= 0, V(X) = α 2 −(E(X)) 2 = α 2 = ϕ′′ (0)
i 2 = 0.
But a random variable with mean and variance equal to zero is a
degenerate variable at zero, that is, P(X = 0) = 1, and then its
c.f. is
ϕ(t) = E [ e it0] = 1, ∀t ∈ IR,
which is a contradiction.
We have seen already that the d.f. determines the c.f. The
following theorem gives an expression of the d.f. in terms of the
c.f. for an interval. This result will imply that the c.f. determines
a unique d.f.
Theorem 1.21 (Inversion Theorem)
Let ϕ(t) be the c.f. corresponding to the d.f. F(x). Let a,b be
two points of continuity of F, that is, a,b ∈ C(F). Then,
F(b)−F(a) = 1
2π lim
T→∞
∫ T
−T
e −ita −e −itb
it
ϕ(t)dt.
As a consequence of the Inversion Theorem, we obtain the following
result.
Theorem 1.22 (The c.f. determines a unique d.f.)
If ϕ(t) is the c.f. of a d.f. F, then it is not the c.f. of any other d.f.
Remark 1.14 (c.f. for an absolutely continuous r.v.)
If F is absolutely continuous with p.d.f. f, then the c.f. is
ϕ(t) = E [ e itX] ∫ ∫
= e itx dF(x) = e itx f(x)dx.
IR
IR
ISABEL MOLINA 49