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 />
Corolary <strong>1.</strong>7 (Properties of a symmetric r.v.)<br />
Let X be a symmetric r.v. Then,<br />
(a) If F X is absolutely continuous, then f X (x) = f X (−x), ∀x ∈<br />
IR.<br />
(b) If F X is discrete, then P X (x) = P X (−x), ∀x ∈ IR.<br />
Theorem <strong>1.</strong>24 (c.f. of a symmetric r.v.)<br />
The c.f. ϕ of the d.f. F is real iff F is symmetric.<br />
Remark <strong>1.</strong>16 We know that ϕ(t), ∀t ∈ IR determines completely<br />
F(x), ∀x ∈ IR. However, if we only know ϕ(t) for t in<br />
a finite interval, then do we know completely F(x), ∀x ∈ IR? The<br />
answer is no, since we can find two different d.f.s with the same c.f.<br />
in a finite interval, see Example <strong>1.</strong>24.<br />
Example <strong>1.</strong>24 (The c.f. in a finite interval does not<br />
determine the d.f.)<br />
Consider the r.v. X taking the values ∓(2n+1), n = 0,1,2,···,<br />
with probabilities<br />
P(X = 2n+1) = P(X = −(2n+1)) =<br />
4<br />
π 2 (2n+1) 2, n = 0,1,2,···<br />
Consideralsother.v. Y takingvalues0,∓(4n+2),n = 0,1,2,...,<br />
with probabilities<br />
P(Y = 0) = 1/2<br />
P(Y = 4n+2) = P(X = −(4n+2)) =<br />
2<br />
π 2 (2n+1) 2, n = 0,1,2,···<br />
ISABEL MOLINA 51