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Unformatted text preview: Derivation of Poisson Density By Jeffrey B. Burl In class, I presented the probability of k calls during dinner: n! a a Pr(k calls during dinner) = lim 1  n k !( n  k )! n n
n k ( nk ) a 1 k  a k n(n  1)L (n  k + 1) a n ak a e = e a = lim 1 . k n nk k! a k! 1 k! 1  n
Looking at the individual limits: (1) n * lim c n k + c1n k 1 + L + n n( n  1)L (n  k + 1) c = lim = lim + 1 + L + kk1 1 ; 1 = k k n n n n n 1 n a lim  = 1k = 1 ; 1 nn n
k Lastly, the one I had trouble with (a tricky little limit): a lim  = 1n , 1 nn n
which is an indeterminate form. This expression can be evaluated: a ln  1 n 1 n a ln  1 n 1 . n n (2) ln  1 n ln  1 a lim  = lim e n = lim e n = e 1 n H * n n n n a n a =e nn lim (3) Using L'Hpital's rule, 1 a a 2 d a a n ln  1 ln  1  1 . n lim dn n lim n = lim  a 1  a lim = = = ( ) a * ... n( n n n 1 d 1 1   2 1 n dn n n n
In conclusion, substituting this into Ex. 3, the limit in Eq. 2 is: a lim  = e  a . 1 n n
Putting this result into Eq. 1, yields the Poisson probability mass function: n Pr(k events during time t) = e  at a k , k! Where a is the probability of an event during unit time. Note that the example presented was for a time of 1 unit, this is why the t appears in this density function. ...
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This note was uploaded on 06/16/2010 for the course EE ee3180 taught by Professor Burl during the Spring '10 term at Michigan Technological University.
 Spring '10
 Burl

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