``` wu :: forums (http://www.ocf.berkeley.edu/~wwu/cgi-bin/yabb/YaBB.cgi) 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 ishttp://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] Powered by YaBB 1 Gold - SP 1.4! Forum software copyright © 2000-2004 Yet another Bulletin Board