Stochastic

More realistic models two of the weaknesses of the

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Unformatted text preview: e t by the independent increments property in (iii), so ⌧2 is exponential( ) and independent of ⌧1 . Repeating this argument we see that ⌧1 , ⌧2 , . . . are independent exponential( ). Up to this point we have been concerned with the mechanics of deﬁning the Poisson process, so the reader may be wondering: Why is the Poisson process important for applications? Our answer is based on the Poisson approximation to the binomial. Suppose that each of the n students on Duke campus ﬂips coins with probability /n of heads to decide if they will go to the Great Hall (food court) between 12:17 and 12:18 . The probability that exactly k students will go during the one-minute time interval is given by the binomial(n, /n) distribution n(n 1) · · · (n k! k + 1) ✓ ◆k ✓ n 1 n ◆n k (2.14) Theorem 2.8. If n is large the binomial(n, /n) distribution is approximately Poisson( ). Proof. Exchanging the numerators of the ﬁrst two fractions and breaking the last term into two, (2.14) becomes k k! · n(n 1) · · · (n nk k + 1) ✓ ·1 n ◆n ✓ 1 n ◆ k (2.15) Considering the four terms sep...
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This document was uploaded on 03/06/2014 for the course MATH 4740 at Cornell.

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