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   Author  Topic: Limit of Integral  (Read 755 times)
ThudnBlunder
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Limit of Integral  
« on: Jul 12th, 2008, 9:11am »		 Quote Quote Modify Modify
What is (1 + t/k)ke-t.dt/k from t = 0 to ?  
           k->
 
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Eigenray
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Re: Limit of Integral  
« Reply #1 on: Jul 12th, 2008, 11:28am »		 Quote Quote Modify Modify
I think it helps to know that the median of Poisson-k is around k.
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Re: Limit of Integral  
« Reply #2 on: Aug 7th, 2008, 10:04am »		 Quote Quote Modify Modify
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By induction we have 0 tre-tdt = r!.  So the integral is
 
r=0k  C(k,r)(k-r)!/kk-r = k! ek/kk r=0k  e-kkr/r!
 = k! ek/kk Pr( Pk k ),
 
where Pk is Poisson-k.  Pk has the same distribution as the sum of k P1's, so by the central limit theorem, (Pk-k)/{k} converges to standard normal, and Pr( Pk k ) converges to 1/2.  By Stirling, k! ek/kk ~ {2k}, and it follows that the limit is {/2}.
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