Talk:Poisson limit theorem

Example review
It's been a little hard for me to understand why this example is an application of the theorem, so I thought I could suggest an extra sentence explaining a intermediate step of reasoning for the not-so-much-into-the-field people like me. Something like:

" Suppose that in an interval [0, 1000], 500 points are placed randomly. Now what is the number $$k$$ of points that will be placed in [0, 10]?

'''Intuitively, it will be most likely $$k=5$$, but other amounts are possible too, so the number $$k$$ should be instead a distribution. More correctly phrased,''' the probabilistically precise way of describing the number of points in the sub-interval would be to describe it as a binomial distribution $$p_n(k)$$. "

As I'm not sure whether it is worth to include such an impreciseness, I'm writing this in the talk page, asking for approval/rejection. But I think the implicit idea (expressed like this or in a better way) would be indeed useful.

jmmut 131.111.184.26 (talk) 17:38, 1 October 2015 (UTC)

Possible typo in proof
The proof's first equation is

\begin{align} {n \choose k} p^k (1-p)^{n-k} &\simeq \lim_{n\to\infty}\frac{n(n-1)(n-2)\dots(n-k+1)}{k!} \left(\frac{\lambda}{n}\right)^k \left(1- \frac{\lambda}{n}\right)^{n-k} \\ &= \lim_{n\to\infty}\frac{n^k+O\left(n^{k-1}\right)}{k!}\frac{\lambda^k}{n^k} \left(1- \frac{\lambda}{n}\right)^{n-k} \\ &= \lim_{n\to\infty}\frac{\lambda^k}{k!} \left(1-\frac{\lambda}{n}\right)^{n-k} \end{align} $$.

but it should be:



\begin{align} \lim_{n\to\infty}{n \choose k} p^k (1-p)^{n-k} &= \lim_{n\to\infty}\frac{n(n-1)(n-2)\dots(n-k+1)}{k!} \left(\frac{\lambda}{n}\right)^k \left(1- \frac{\lambda}{n}\right)^{n-k} \\ &\simeq \lim_{n\to\infty}\frac{n^k+O\left(n^{k-1}\right)}{k!}\frac{\lambda^k}{n^k} \left(1- \frac{\lambda}{n}\right)^{n-k} \\ &= \lim_{n\to\infty}\frac{\lambda^k}{k!} \left(1-\frac{\lambda}{n}\right)^{n-k} \end{align} $$.

--Corrado Mencar (talk) 14:11, 28 April 2020 (UTC)