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$\lim_{n\rightarrow \infty} \sum_{i=0}^{n}\frac{e^{-n}n^{i}}{i!}\rightarrow \frac{1}{2}$

Tried: here suppose N is poission distribution with parameter n $\lim_{n\rightarrow \infty} \sum_{i=0}^{n} P(N\geq i) $ $= \lim_{n\rightarrow \infty} \sum_{i=0}^{n}(P(N=i)+P(N>i) + P(N<i)-P(N<i)) $ $= \lim_{n\rightarrow \infty} \sum_{i=0}^{n}(P(N= i)+P(i<N<i)-P(N<i)$ $= \lim_{n\rightarrow \infty} \sum_{i=0}^{n}(P(N= i)-P(N<i)$ ; $(P(i<N<i)=0)$ $=\sum_{i=0}^{\infty}(P(N= i)-\sum_{i=0}^{\infty}P(N<i)$ $=1-\frac{1}{2}$ $=\frac{1}{2}$

Ashutosh Kumar Maurya
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  • What you've written makes no sense. You wanted $P(N \leq i)$. If you imitated your first step and then carried through the remaining steps correctly, you would arrive at $1-P(N> i)$, which is useless. Anyway, your expression is $P(X_n \leq n)$ where $n$ has the Poisson(n) distribution. There is a famous way to approximate the Poisson(n) distribution for large $n$... – Ian May 24 '16 at 13:58
  • I am not getting ... – Ashutosh Kumar Maurya May 24 '16 at 14:02
  • The Poisson(n) distribution can be realized by summing n independent, identically distributed Poisson(1) variables. This "infinite divisibility" property is one of the basic properties of the Poisson distribution (along with several other famous distributions...) – Ian May 24 '16 at 14:03
  • By the law of large numbers, the probability that a limit Poisson distribution is less than its mean value is $\frac{1}{2}$. – Jack D'Aurizio May 24 '16 at 14:26
  • @JackD'Aurizio Actually, it's the central limit theorem, not the law of large numbers. The weak law of large numbers only tells you that $P((1-\varepsilon)n \leq X_n \leq (1+\varepsilon)n)$ goes to $1$ for any fixed $\varepsilon > 0$. It doesn't say how $P((1-\varepsilon)n \leq X_n < n)$ compares to $P(n < X_n \leq (1+\varepsilon)n$. – Ian May 24 '16 at 14:50
  • @Ian: I apologize, you're right. Anyway, I think this is a duplicate. – Jack D'Aurizio May 24 '16 at 14:51
  • @JackD'Aurizio Yes, it is, it's not that hard to find it, but I'd like to see Ashutosh think about it a bit. With some thought, there is some very interesting structure here in the connection between the binomial, Poisson, and normal distributions. Being told the detailed answer makes it seem like "just another problem" with no real meaning. – Ian May 24 '16 at 14:52

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