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parameter p. Therefore, Nt has the binomial distribution with parameters t/h and p = λh. So
E [Nt ] = t/h hλ ≈ λt. Recall from Section 2.7 that the limit of a binomial distribution as n → ∞
and p → 0 with np → λ is the Poisson distribution with parameter λ. Therefore, as h → 0, the
limiting distribution of Nt is the Poisson distribution with mean λt. More generally, if 0 ≤ s < t,
the distribution of the increment Nt − Ns converges to the Poisson distribution with parameter
(t − s)λ. Also, the increments of Nt over disjoint intervals are independent random variables. 3.5.2 Deﬁnition and properties of Poisson processes A Poisson process with rate λ > 0 is obtained as the limit of scaled Bernoulli random counting
processes as h → 0 and p → 0 such that p/h → λ. This limiting picture is just used to motivate the
deﬁnition of Poisson processes, given below, and to explain why Poisson processes naturally arise
in applications. A sample path of a Poisson process is shown in Figure 3.8. The variable N (t) for
T1 U2 U3 ... T2 T3 Figure 3.8: A sample path of a Poisson process...
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This note was uploaded on 02/09/2014 for the course ISYE 2027 taught by Professor Zahrn during the Spring '08 term at Georgia Tech.
- Spring '08
- The Land