Originally Posted by

**Mathsdog** I wonder if someone could clarify these limits for me. They arise in the derivation of the poisson distribution from the binomial distribution as $\displaystyle \lim_{ n \to +\infty}$ and $\displaystyle \lim_{ p \to 0}$

The first is

$\displaystyle [1- (\lambda/n)]^{n} \rightarrow \mathrm{e}^{-\lambda}$

as $\displaystyle n \to +\infty$.

I don't see how this works. If anything I would have thought that

$\displaystyle (1- (\lambda/n))^{n} = \mathrm{e}^{n \cdot \log{(1-\lambda/n)}} \to \mathrm{e}^{0} = 1 $

as $\displaystyle n \to +\infty$.

The second is that

\frac{n!}{(n-k)!(n-\lambda)^k = \frac{n(n-1) \ldots (n-k+1)}{(n - \lambda) (n-\lambda) \ldots (n - \lambda)}

is a "quantity that tends to 1 as $\displaystyle n \to +\infty$ (since $\displaystyle \lambda$ remains constant)."

What precisely is the role of $\displaystyle \lambda$? Is the critical condition not that the numerator and denominator grow at the same rate, and if so, how can this be demonstrated?

Thanks in advance. MD