Poisson Processes
To prepare for our discussion of the Poisson process, we need to recall the definition and some of the basic properties of the exponential distribution. A random variable T is said to have an exponential distribution with rate λ, or T = exponential(λ), if
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Poisson Processes
2.1 Exponential Distribution To prepare for our discussion of the Poisson process, we need to recall the definition and some of the basic properties of the exponential distribution. A random variable T is said to have an exponential distribution with rate , or T D exponential(), if P.T t/ D 1 et
for all t 0
(2.1)
Here we have described the distribution by giving the distribution function F.t/ D P.T t/. We can also write the definition in terms of the density function fT .t/ which is the derivative of the distribution function. ( et for t 0 (2.2) fT .t/ D 0 for t < 0 Integrating by parts with f .t/ D t and g0 .t/ D et , Z ET D
1
0
t et dt
ˇ1 D tet ˇ0 C
Z
1 0
et dt D 1=
(2.3)
Integrating by parts with f .t/ D t2 and g0 .t/ D et , we see that ET 2 D
Z 0
1
t2 et dt ˇ
1 t2 et ˇ0
Z
1
2tet dt D 2=2
(2.4)
© Springer International Publishing Switzerland 2016 R. Durrett, Essentials of Stochastic Processes, Springer Texts in Statistics, DOI 10.1007/978-3-319-45614-0_2
95
D
C 0
96
2 Poisson Processes
by the formula for ET. So the variance var .T/ D ET 2 .ET/2 D 1=2
(2.5)
While calculus is required to know the exact values of the mean and variance, it is easy to see how they depend on . Let T D exponential./, i.e., have an exponential distribution with rate , and let S D exponential.1/. To see that S= has the same distribution as T, we use (2.1) to conclude P.S= t/ D P.S t/ D 1 et D P.T t/ Recalling that if c is any number then E.cX/ D cEX and var .cX/ D c2 var .X/, we see that ET D ES=
var .T/ D var .S/=2
Lack of Memory Property It is traditional to formulate this property in terms of waiting for an unreliable bus driver. In words, “if we’ve been waiting for t units of time then the probability we must wait s more units of time is the same as if we haven’t waited at all.” In symbols P.T > t C sjT > t/ D P.T > s/
(2.6)
To prove this we recall that if B A, then P.BjA/ D P.B/=P.A/, so P.T > t C sjT > t/ D
e.tCs/ P.T > t C s/ D D es D P.T > s/ P.T > t/ et
where in the third step we have used the fact eaCb D ea eb . Exponential Races Let S D exponential() and T D exponential() be independent. In order for the minimum of S and T to be larger than t, each of S and T must be larger than t. Using this and independence we have P.min.S; T/ > t/ D P.S > t; T > t/ D P.S > t/P.T > t/ D et et D e.C/t
(2.7)
That is, min.S; T/ has an exponential distribution with rate C . We will now consider: “Who finishes first?” Breaking things down according to the value of S and then using independence with our formulas (2.1) and (2.2) for the distribution and density functions, to conclude Z P.S < T/ D
0
1
fS .s/P.T > s/ ds
2.1 Exponential Distribution
Z D 0
D
97 1
es es ds
C
Z
1 0
. C /e.C/s ds D
C
(2.8)
where on the last line we have used the fact that .C/e.C/s is a density function and hence must integrate to 1. Example 2.1. Anne and Betty enter a beauty parlor simultaneou
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