Slides Chapter 1. Measure Theory and Probability

1.9. THE CHARACTERISTIC FUNCTION
Using the formulas
∞∑ 1
(2n+1) = π2
2 8 ;
prove:
n=0
8 ∑ ∞
π 2
n=0
(a) P X is a probability function;
(b) P Y is a probability function;
(c) ϕ X (t) = ϕ Y (t), for |t| ≤ π/2.
cos(2n+1)t
= 1− 2|t| , |t| ≤ π,
(2n+1) 2 π
Remark 1.17 From the series expansion of the c.f. in Corollary
1.4, one is tempted to conclude that the c.f., and therefore also
the d.f., of a r.v. is completely determined by all of its moments,
provided that they exist. This is false, see Example 1.25.
Example 1.25 (Moments do not always determine the
c.f.)
For a ∈ [0,1], consider the p.d.f. defines by
f a (x) = 1 24 e−x1/4 (1−asin(x 1/4 ), x ≥ 0.
Using the following formulas:
prove:
∫ ∞
∫0
∞
0
(a) f a is a p.d.f. ∀a ∈ [0,1].
x n e −x1/4 sin(x 1/4 )dx = 0, ∀n ∈ IN ∪{0};
x n e −x1/4 = 4(4n+3)!∀n ∈ IN ∪{0},
ISABEL MOLINA 52