that X(t)=ri and suppose there is a matrix-valued function r (t) such that u(t) satisfies t)=r(t)(t) Then we call X a finite-state continuous time Markov chain and If r(t)=r we call the process and time-homogenenous or stationary 3.3 Poisson processes 3.3.1 The poisson distribution Intuitively, the Poisson distribution comes from taking the limit of a sum of bernoulli random var tables Definition, a random variable x is said to be bernoulli if there is a real number 0≤p≤1 such that P Definition. A random variable y is said to be binomial distribution if there is an integer n and a real number0≤p≤1 such that P(Y=6)=∑(x)少(-p where the binomial coefficient is defined as followsthat X(t) = xi and suppose there is a matrix-valued function Γ(t) such that µ(t) satisfies µ˙(t) = Γ(t)µ(t). Then we call X a finite-state continuous time Markov chain and If Γ(t) = Γ we call the process and time–homogenenous or stationary. 3.3 Poisson processes 3.3.1 The Poisson distribution Intuitively, the Poisson distribution comes from taking the limit of a sum of Bernoulli random variables. Definition. A random variable X is said to be Bernoulli if there is a real number 0 ≤ p ≤ 1 such that P({X = 1}) = p and P({X = 0}) = 1 − p. Definition. A random variable Y is said to be binomial distribution if there is an integer n and a real number 0 ≤ p ≤ 1 such that P({Y = k}) = Xn k=0 µ n k ¶ p k (1 − p) n−k where the binomial coefficient is defined as follows. µ n k ¶ = n! k!(n − k)! 18