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

A.3 Probability TheoryDefinition A.4 (Continuous valued Random Variables and Mathematical Expectation withdensities).Consider a function f defined on R 1 = (−∞,+∞):u → f (u) with u ∈ R 1 ,and satisfying the two conditions:1. u → f (u) ≥ 0,2.´ +∞−∞f (u)du = 1.Such a function is called a density function on R 1 .We are now able to define also the probabilities for any continuous valued random variable X bymeans of a density function f so that for any closed interval [a,b] we haveP(a ≤ X ≤ b) = ´ baf (u)du.More generally, if A is the union of intervals not necessarily disjoint and some of which may beinfinite, we haveP(X ∈ A) =´Af (u)du.The mathematical expectation for X and any arbitrary function of it, Φ(X), then is defined asfollows:E(X) =E(Φ(X)) =ˆ +∞−∞ˆ +∞−∞f (u)du,Φ(u) f (u)du.Definition A.5 (Conditional Probability).Let A,B ⊂ Ω be two events. Then P(A|B) is denoted as the probability of event A suchthat B has occured and is defined as:P(A|B) = P(A∩B)P(B), if P(B) ≠ 0.103