We need to find out the limit of,
limn→∞∑nk=0e−nnkk!
One can see that e−nnkk! is the cdf of Poisson distribution with parameter n.
Please give some hints on how to find out the limit.
Answer
It's a good start to try to solve it in a probabilistic way: notice that the Poisson random variable has the reproducibility property, that is, if Xk∼Poisson(1), k=1,2,…,n independently, then
Sn=n∑k=1Xk∼Poisson(n),
whose distribution function FSn satisfies:
FSn(n)=P[Sn≤n]=n∑k=0e−nnkk!,
which is exactly the expression of interest. Hence this suggests linking this problem to central limit theorem.
By the classic CLT, we have
Sn−n√n⇒N(0,1).
Hence
P[Sn≤n]=P[Sn−n√n≤0]→P[Z≤0]=12
as n→∞.
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