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riddles >> putnam exam (pure math) >> Limit of Integral
(Message started by: ThudanBlunder on Jul 12th, 2008, 9:11am)

Title: Limit of Integral
Post by ThudanBlunder on Jul 12th, 2008, 9:11am
What is http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/frakcl.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/fraki.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/frakm.gifhttp://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif(1 + t/k)ke-t.dt/http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gifk from t = 0 to http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif?
                                             k->http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/infty.gif


Title: Re: Limit of Integral
Post by Eigenray on Jul 12th, 2008, 11:28am
I think it helps to know that [hide]the median of Poisson-k is around k[/hide].

Title: Re: Limit of Integral
Post by Eigenray on Aug 7th, 2008, 10:04am
[hideb]By induction we have http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/int.gif0http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/supinfty.gif tre-tdt = r!.  So the integral is

http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gifr=0k  C(k,r)(k-r)!/kk-r = k! ek/kk http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gifr=0k  e-kkr/r!
= k! ek/kk Pr( Pk http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gif 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)/http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif{k} converges to standard normal, and Pr( Pk http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gif k ) converges to 1/2.  By Stirling, k! ek/kk ~ http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif{2http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gifk}, and it follows that the limit is http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif{http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/pi.gif/2}.[/hideb]



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