Stat 5101 Lecture Notes - School of Statistics

122 Stat 5101 (Geyer) Course Notesbecause the measure of the union of disjoint regions is the sum of the measures.This is also obvious from linearity of expectation. We must haveE(X 1 + ···+X n )=E(X 1 )+···+E(X n ).Corollary 4.6. The conditional distribution of a Poisson process in a regionA c given the process in A is the same as the unconditional distribution of theprocess in A c .In other words, finding the point pattern in A tells you nothing whatsoeverabout the pattern in A c . The pattern in A c has the same distributionconditionally or unconditionally.Proof. By Definition 4.4.1 and Theorem 4.4 N B is independent of N C whenB ⊂ A c and C ⊂ A. Since this is true for all such C, the random variable N Bis independent of the whole pattern in A, and its conditional distribution giventhe pattern in A is the same as its unconditional distribution. Theorem 4.4 saysPoisson distributions of the N B for all subsets B of A c imply that the processin A c is a Poisson process.4.4.3 One-Dimensional Poisson ProcessesIn this section we consider Poisson processes in one-dimensional space, thatis, on the real line. So a realization of the process is a pattern of points on theline. For specificity, we will call the dimension along the line “time” because formany applications it is time. For example, the calls arriving at a telephone exchangeare often modeled by a Poisson process. So are the arrivals of customersat a bank teller’s window, or at a toll plaza on an toll road. But you shouldremember that there is nothing in the theory specific to time. The theory is thesame for all one-dimensional Poisson processes.Continuing the time metaphor, the points of the process will always in therest of this section be called arrivals. The time from a fixed point to the nextarrival is called the waiting time until the arrival.The special case of the gamma distribution with shape parameter one iscalled the exponential distribution, denoted Exp(λ). Its density isf(x) =λe −λx , x > 0. (4.10)Theorem 4.7. The distribution of the waiting time in a homogeneous Poissonprocess with rate parameter λ is Exp(λ). The distribution is the same unconditionally,or conditional on the past history up to and including the time westart waiting.Call the waiting time X and the point where we start waiting a. Fix anx>0, let A =(a, a + x), and let Y = N (a,a+x) be the number of arrivals in theinterval A. Then Y has a Poisson distribution with mean λm(A) =λx, since