Diffusion Processes with Hidden States from ... - FU Berlin, FB MI

A AppendixA.3 Probability TheoryDefinition A.1 (Probability Measure).A probability measure P on the sample space Ω is a function of subsets of Ω satisfyingthree axioms:1. For every set A ⊂ Ω, the value of the function is a nonnegative number:P(A) ≥ 0.2. For any two disjoint sets A and B, the value of the function for their union A + B is equal tothe sum of its value for A and B seperately:P(A + B) = P(A) + P(B), provided that AB = ∅.3. Axiom of countable additivity:For a countably infinite collection of disjoint sets A k , k = 1,2,..., we haveP(∑ ∞ k=1A k ) = ∑ ∞ k=1P(A k ).4. The value of the function for Ω (as a subset) is equal to 1:P(Ω) = 1.Definition A.2 (Random Variable).A numerically valued function X of ω with domain Ω:ω ∈ Ω : ω → X(ω)is called a random variable.Definition A.3 (Mathematical Expectation).For a random variable X defined on a countable sample space Ω, its mathematical expectationis the number E(X), given by the formula:E(X) = ∑ ω∈Ω X(ω)P({ω}),provided that the series converges absolutely, namely∑ ω∈Ω |X(ω)|P({ω}) < ∞.In this case we say that the mathematical expectation of X exists.102