Using Central Limit Theorem

Question

Can anyone help me with it: Using the central limit theorem for suitable Poisson random variables, prove that $$ \lim_{n\to\infty} e^{-n} \sum_{k=0}^{n} \frac{n^k}{k!}=1/2$$ Thanks!

This question has been asked and answered countless times. – Lucian Dec 07 '15 at 21:50 — Lucian, Dec 07 '15 at 21:50

score 10 · Accepted Answer · answered Feb 13 '11 at 20:44

10

Hint: A Poisson$(n)$ random variable can be represented as the sum of $n$ i.i.d. Poisson$(1)$ rv's.

answered Feb 13 '11 at 20:44

Shai Covo

24,077

I'll give further hints if you wish. – Shai Covo Feb 13 '11 at 20:54
@Shai Covo: Can you show me the steps? Thanks! – kira Feb 13 '11 at 21:55
1

@kira: Hint 2: Move the exponent into the sum. – Shai Covo Feb 13 '11 at 21:59
1

@kira: Find a random variable $X$ for which it holds ${\rm P}(X=n) = \sum\nolimits_{k = 0}^n {\frac{{e^{ - n} n^k }}{{k!}}} $. – Shai Covo Feb 14 '11 at 10:11

Using Central Limit Theorem

1 Answers1