Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
Slides Chapter 1. Measure Theory and Probability
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<strong>1.</strong>9. THE CHARACTERISTIC FUNCTION<br />
We have seen that for absolutely continuous F, the c.f. ϕ(t) can<br />
be expressed in terms of the p.d.f. f. However, it is possible to<br />
express the p.d.f. f in terms of the c.f. ϕ(t)? The next theorem is<br />
the answer.<br />
Theorem <strong>1.</strong>23 (Fourier transform of the c.f.)<br />
If F is absolutely continuous <strong>and</strong> ϕ(t) is Riemann integrable in IR,<br />
that is, ∫ ∞<br />
−∞<br />
|ϕ(t)|dt < ∞, then ϕ(t) is the c.f. corresponding to<br />
an absolutely continuous r.v. with p.d.f. given by<br />
f(x) = F ′ (x) = 1 ∫<br />
e −itx ϕ(t)dt,<br />
2π<br />
where the last term is called the Fourier transform of ϕ.<br />
In the following, we are going to study the c.f. of r<strong>and</strong>om variables<br />
that share the probability symmetrically in IR + <strong>and</strong> IR − .<br />
Definition <strong>1.</strong>52 (Symmetric r.v.)<br />
A r.v. X is symmetric if <strong>and</strong> only if X d = −X, that is, iff F X (x) =<br />
F −X (x), ∀x ∈ IR.<br />
Remark <strong>1.</strong>15 (Symmetric r.v.)<br />
Since ϕ is determined by F, X is symmetric iff ϕ X (t) = ϕ −X (t),<br />
∀t ∈ IR.<br />
Corolary <strong>1.</strong>5 (Another characterization of a symmetric<br />
r.v.)<br />
X is symmetric iff F X (−x) = 1 − F X (x − ), where F X (x − ) =<br />
P(X < x).<br />
Corolary <strong>1.</strong>6 (Another characterization of a symmetric<br />
r.v.)<br />
X is symmetric iff ϕ X (t) = ϕ X (−t), ∀t ∈ IR.<br />
IR<br />
ISABEL MOLINA 50