Slides Chapter 1. Measure Theory and Probability

1.9. THE CHARACTERISTIC FUNCTION
Corolary 1.7 (Properties of a symmetric r.v.)
Let X be a symmetric r.v. Then,
(a) If F X is absolutely continuous, then f X (x) = f X (−x), ∀x ∈
IR.
(b) If F X is discrete, then P X (x) = P X (−x), ∀x ∈ IR.
Theorem 1.24 (c.f. of a symmetric r.v.)
The c.f. ϕ of the d.f. F is real iff F is symmetric.
Remark 1.16 We know that ϕ(t), ∀t ∈ IR determines completely
F(x), ∀x ∈ IR. However, if we only know ϕ(t) for t in
a finite interval, then do we know completely F(x), ∀x ∈ IR? The
answer is no, since we can find two different d.f.s with the same c.f.
in a finite interval, see Example 1.24.
Example 1.24 (The c.f. in a finite interval does not
determine the d.f.)
Consider the r.v. X taking the values ∓(2n+1), n = 0,1,2,···,
with probabilities
P(X = 2n+1) = P(X = −(2n+1)) =
4
π 2 (2n+1) 2, n = 0,1,2,···
Consideralsother.v. Y takingvalues0,∓(4n+2),n = 0,1,2,...,
with probabilities
P(Y = 0) = 1/2
P(Y = 4n+2) = P(X = −(4n+2)) =
2
π 2 (2n+1) 2, n = 0,1,2,···
ISABEL MOLINA 51