Networks and Random Processes, MathSys MA933, 2016
نویسندگان
چکیده
(a) Consider a sequence of independent coin tosses with P[heads] = p and let Yn be the number of heads up to time n with Y0 = 0. Give the state space S and transition matrix for the process (Yn : n ∈ N0). Compute P[Yn = k] for all n, k ≥ 0 using the binomial formula. (b) Let (Zn : n ∈ N0) be a simple random walk on S = Z with transition matrix p(x, y) = pδy,x+1 + qδy,x−1. Use that Zn = 2Yn − n to get a formula for P[Zn = k] for all n, k ≥ 0. Does (Zn : n ∈ N0) have a stationary distribution on Z? (c) Stirling’s formula says that
منابع مشابه
MA933 - Networks and Random Processes MSc in Mathematics of Systems
Definition 1.1 A probability distribution P on (Ω,F) is a function P : F → [0, 1] which is (i) normalized, i.e. P[∅] = 0 and P[Ω] = 1 (ii) additive, i.e. P [ ∪i Ai ] = ∑ i P[Ai] , where A1,A2, . . . is a collection of disjoint events, i.e. Ai ∩ Aj = ∅ for all i, j. The triple (Ω,F ,P) is called a probability space. For discrete Ω: F = P(Ω) and P[A] = ∑ω∈A P[ω] e.g. P[even number on a die] = P[2...
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