Chapter 9 Poisson processes - Yale University

1 Page 1 Chapter 9 Poisson processesThe Binomial distribution and the geometric distribution describe the behavior of tworandom variables derived from the random mechanism that I have called coin tossing . Thenamecoin tossingdescribes the whole mechanism; the namesBinomialandgeometricreferto particular aspects of that mechanism. If we increase the tossing rate to m tosses per sec-ond and decrease the probability of heads to a small p, while keeping the expected numberof heads per second fixed at Dmp, the number of heads in atsecond interval will haveapproximately a ;p/distribution, which is close to the Poisson .

2 T/. Also, the num-bers of heads tossed during disjoint time intervals will still be independent random the limit, asm!1, we get an idealization called aPoisson process. Poisson process< > Poisson process with rate on[0;1/is a random mechanism that gener-ates points strung out along[0;1/in such a way that(i) the number of points landing in any subinterval of lengthtis a random variable witha Poisson . t/distribution(ii) the numbers of points landing in disjoint (= non-overlapping) intervals are indepen-dent random variables. The double use of the name Poisson is unfortunate.]]

3 Much confusion would be avoidedif we all agreed to refer to the mechanism as idealized-very-fast-coin-tossing , or somesuch. Then the Poisson distribution would have the same relationship to idealized-very-fast-coin-tossing as the Binomial distribution has to coin-tossing. Obversely, we could createmore confusion by renaming coin tossing as the binomial process . Neither suggestion islikely to be adopted, so you should just get used to having two closely related objects withthe name bother about Poisson processes ? When we pass to the idealized mechanism ofpoints generated in continuous time, several awkward artifacts of discrete-time coin tossingdisappear.

4 The Examples and Exercises in this Chapter will illustrate the simplifications.< > the distribution of the time to thekth point in a Poisson process on [0;1/with rate .Solution:Denote the time to thekth point byTk. It has a continuous distribution, whichis specified by a density function. Fort>0 and small >0,Pft Tk<tC gDPfexactlyk 1 points in [0;t/, exactly one point in [t;tC /gCsmaller order termsThe smaller order terms contribute probability less thanPf2 or more points in [t;tC /gDPfPoisson. / 2gDe . /22!C:::Statistics 241: 28 October 1997c David PollardChapter 9 Poisson processesPage 2By the independence property (ii) for Poisson processes , the main term factorizes asPfexactlyk 1 points in [0;t/gPfexactly one point in [t;tC /gDe t.]]]]]]

5 T/k 1/!e t. /11!De t ktk 1 .k 1/!Csmaller order termsThat is, the distribution ofTkhas densitye t ktk 1/!fort>0: Gamma function and gamma densityIt is easier to remember the form of the density forTkif one rescales, using an argumentYou should try thiscalculation at to the one for theN. ; 2/distribution in Chapter 7, to show that Tkhas a distri-bution with density< >e ttk 1/!fort>0:This density is called generally, for each >0, the densitye tt 10. /fort>0is called thegamma. /density. The scaling constant,0. /, which ensures that the den- gamma.

6 /densitysity integrates to one, is given by0. /DZ10e xx 1dxfor each >0:The function0. /is called thegamma function. Don t confuse the gamma density with gamma functionthe gamma function.< > waiting timeTkfrom Example< >has expected valueETkDZ10te t ktk 1/!dtD1 Z10e 1/!dxputtingxD t, (cf. distribution of Tk)Dk (Use integration by parts.)Does it make sense to you thatETkshould decrease as increases?More generally, for >0,0. C1/DZ10e xx dxD e xx 10C Z10e xx 1dxD 0. /In particular, 1 1 1/.k 2 2/D:::Statistics 241: 28 October 1997c David PollardChapter 9 Poisson processesPage 1/.

7 K 2/.k 3/:::.2/.1 1/! xdxD1. Compare with the fact that the in< >integrates to one. Exponential distributionSpecializing the the casekD1 we get the densitye tfort>0;which is called the (standard)exponential distribution. The time to the first point in exponential distributionthe Poisson process has density e tfort>0;an exponential distribution with expected value 1= . Don t confuse the exponential densitywith the exponential the parallels between the negative binomial distribution (in discrete time) and thegamma distribution (in continuous time).

8 Each distribution corresponds to the waiting timeto thekth occurrence of something, for various values ofk. Just as (see Problem Sheet 4)the negative binomial can be written as a sum of independent random variables, each with ageometric distribution, so can the written as a sum ofkindependent randomcts. time$discrete timegamma$neg. binomialexponential$geometricFor counts: Poisson $Binomialvariables, each with an exponential distribution. The times between points in a Poisson pro-cess are independent, exponentially distributed, random gamma distribution turns up in a few unexpected places.

9 < > a standard normal distribution, with density .t/Dexp. t2=2/=p2 for 1<t<1, show thatZ2=2 has a gamma(1/2) :WriteYforZ2=2. It has a continuous distribution concentrated on the posi-tive half ;1/.Fory>0, and >0 small,Pfy<Y<yC gDPf2y<Z2<2yC2 gDPfp2y<Z<p2yC2 or p2yC2 <Z< p2ygNotice the two contributions; the square function is not one-to-one. Students who memorizeand blindly apply transformation formulae quite often overlook such multiple gives a good approximation to the length of the short interval fromp2ytop2yC2 . Temporarily Thenp2yC2 / =p2yThe interval from p2yC2 to p2yhas the same length.

10 Using the approximationPfx<Z<xC g .x/for small >0;deduce thatPfy<Y<yC g p2y .p2y/C p2y . p2y/D2 p2y1p2 exp p2y 2=2 D p y 1=2e yThat is,Yhas the distribution with density1p y 1=2e yfory>0:Statistics 241: 28 October 1997c David PollardChapter 9 Poisson processesPage 4 Compare with the ,y1 1=2e >0:The distribution ofZ2=2 is gamma (1/2), as : From the fact that the density must integrate to 1, we get a 1e ydyDp Actually, you could arrive at the same conclusion by making the change of variableyDx2=2 in the integral which is effectively what we have done in finding the density for therandom variableZ2=2.