Transcription of Chapter 1 Poisson Processes - NYU Courant
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Chapter 1 Poisson Processes - NYU Courant
Chapter 1 Poisson The Basic Poisson ProcessThe Poisson Process is basically a counting processs. A Poisson Process onthe interval [0, ) counts the number of times some primitive event hasoccurred during the time interval [0, t]. The following assumptions are madeabout the Process N(t).(i). The distribution ofN(t+h) N(t) is the same for eachh >0, isindependent oft.(ii). The random variablesN(t j) N(tj) are mutually independent if theintervals [tj, t j] are nonoverlapping.(iii).N(0) = 0,N(t) is integer valued, right continuous and nondecreasingint, with Probability 1.]
2 CHAPTER 1. POISSON PROCESSES (3). The gaps τ1,τ2,··· between successive jumps are independent identically distributed random variables with the exponential distribution P{τj ≥ x} = (exp[−λx] for x ≥ 0 1 for x ≤ 0 (1.1) Proof. Let us divide the interval [0,T] into n equal parts and compute the expected number of intervals with N ...
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