Probability on Graphs Summary of Lectures

Let Ω be a sample space. We are going to assume that Ω is either a finite set {ω1, . . . , ωn}, or infinite countable set {ωi, i ∈ N}, where N = {1, . . . , } is the set of positive integers. The two kinds of such sets are called countable. Denote by #Ω the cardinality of Ω. Thus #Ω = n ∈ N if Ω is a finite set. #Ω = א0 if Ω is infinite countable. To each ω ∈ Ω we attach a probability (mass) p(ω) ≥ 0. The normalization condition is ω∈Ω p(ω) = 1. That is, the total mass of Ω is 1. The row vector μ := (p(ω1), p(ω2), . . .) is also called sometimes a probability measure on Ω, or distribution. A subset A of Ω, denoted as A ⊂ Ω is called an event. The set of all subsets of Ω is denoted by 2. It includes the empty set, denoted by ∅ and Ω. Then Pr(A) := ω∈Ω p(ω) is the probability of the event A. It is agreed the Pr(∅) = 0. Clearly Pr(Ω) = 1. Example. Assume that Ω = {ω1, . . . , ωn} is a finite space. Let p(ω) = 1 n for any ω ∈ Ω. Then Pr(A) = #A #Ω . (Note #A = 0 ⇐⇒ A = ∅.) Such probability is called uniform distribution. Let A,B ∈ 2 be two events. Then A ∩ B is the intersection of A and B, is the set which consists of all elements which belong to A and B. A ∪B is the union of A and B, is the set which consists of all elements which belong either to A and to B. Then