From: Robin Chapman <rjc@maths.ex.ac.uk>
Subject: Re: exp(-n) sum_{t=0}^{n-1} n^t/t! -> 1/2 (?)
Date: Mon, 21 Aug 2000 15:23:08 GMT
Newsgroups: sci.math.research
Summary: [missing]

In article <8nr854$i9k$1@esel.cosy.sbg.ac.at>,
  gwesp@cosy.sbg.ac.at wrote:
> numerical evidence suggests that the above should tend to 1/2 as n tends
> to infinity.  i'm unable to prove this fact, however.  i'd greatly
> appreciate if somebody in the group has any ideas!
>
Apply the Central Limit Theorem. What you have is the probability
that X_n < E(X_n) where X_n is a Poisson random variable with mean n.
Now X_n has the same distribution as a sum of n independent mean 1
Poisson variables. It follows by the Central Limit Theorem that
if Y is a random variable with mean and variance, and X_n is
the sum of n indepdent variables with the distribution of Y, then
P(X_n < E(X_n)) --> 1/2 as n --> infinity.

--
Robin Chapman, http://www.maths.ex.ac.uk/~rjc/rjc.html
 "`The twenty-first century didn't begin until a minute
  past midnight January first 2001.'"
   John Brunner, _Stand on Zanzibar_ (1968)


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