Slides Chapter 1. Measure Theory and Probability

1.9. THE CHARACTERISTIC FUNCTION
Theorem 1.17 (Markov’s inequality)
Let X be a r.v. from (Ω,A,P) in (IR,B) and g be a non-negative
r.v. from (IR,B,P X ) in (IR,B) and let k > 0. Then, it holds
P ({ω ∈ Ω : g(X(ω)) ≥ k}) ≤ E[g(X)] .
k
Theorem 1.18 (Tchebychev’s inequality)
Let X be a r.v. with finite mean µ and finite standard deviation
σ. Then
P ({ω ∈ Ω : |X(ω)−µ| ≥ kσ}) ≤ 1 k 2.
Corolary 1.3 Let X be a r.v. with mean µ and standard deviation
σ = 0. Then, P(X = µ) = 1.
Problems
1. Prove Markov’s inequality.
2. Prove Tchebychev’s inequality.
3. Prove Corollary 1.3.
1.9 The characteristic function
Wearegoingtodefinethecharacteristicfunctionassociatedwitha
distribution function (or with a random variable). This function is
prettyusefulduetoitscloserelationwiththed.f. andthemoments
of a r.v.
Definition 1.51 (Characteristic function)
Let X be a r.v. defined from the measure space (Ω,A,P) into
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