Derivation%20of%20Poisson%20Density

Derivation%20of%20Poisson%20Density - Derivation of Poisson...

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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 ( n-k ) 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 + kk-1 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.

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