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 oneminute
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.
 Spring '10
 DURRETT
 The Land

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