Stat 5101 Lecture Notes - School of Statistics

4.4. THE POISSON PROCESS 123measure in one dimension is length. Then the c. d. f. of X is given byF (x) =P(X≤x)=P(there is at least one arrival in (a, a + x))= P (Y ≥ 1)=1−P(Y =0)=1−e −λxDifferentiating gives the density (4.10).The assertion about the conditional and unconditional distributions beingthe same is just the fact that the process on (−∞,a] is independent of theprocess on (a, +∞). Hence the waiting time distribution is the same whetheror not we condition on the point pattern in (−∞,a].The length of time between two consecutive arrivals is called the interarrivaltime. Theorem 4.7 also gives the distribution of the interarrival times, becauseit says the distribution is the same whether or not we condition on there beingan arrival at the time we start waiting. Finally, the theorem says an interarrivaltime is independent of any past interarrival times. Since independence is asymmetric property (X is independent of Y if and only if Y is independent ofX), this means all interarrival times are independent.This means we can think of a one-dimensional Poisson process two differentways.• The number of arrivals in disjoint intervals are independent Poisson randomvariables. The number of arrivals in an interval of length t is Poi(λt).• Starting at an arbitrary point (say time zero), the waiting time to thefirst arrival is Exp(λ). Then all the successive interarrival times are alsoExp(λ). And all the interarrival times are independent of each other andthe waiting time to the first arrival.Thus if X 1 , X 2 , ... are i. i. d. Exp(λ) random variables, the times T 1 , T 2 ,... defined byn∑T n = X i (4.11)form a Poisson process on (0, ∞).Note that by the addition rule for the gamma distribution, the time of thenth arrival is the sum of n i. i. d. Gam(1,λ) random variables and hence has aGam(n, λ) distribution.These two ways of thinking give us a c. d. f. for the Gam(n, λ) distributioni=1