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

Title: Limit of Integral
Post by ThudanBlunder on Jul 12^th, 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 12^th, 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 7^th, 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 t^re^-tdt = r!. So the integral is

http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gif_r=0^k C(k,r)(k-r)!/k^k-r = k! e^k/k^k http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/sum.gif_r=0^k e^-kk^r/r!
= k! e^k/k^k Pr( P_k http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gif k ),

where P_k is Poisson-k. P_k has the same distribution as the sum of k P₁'s, so by the central limit theorem, (P_k-k)/http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/surd.gif{k} converges to standard normal, and Pr( P_k http://www.ocf.berkeley.edu/~wwu/YaBBImages/symbols/le.gif k ) converges to 1/2. By Stirling, k! e^k/k^k ~ 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]