I was looking at the web page containing a derivation for the Poisson distribution:(adsbygoogle = window.adsbygoogle || []).push({});

http://en.wikipedia.org/wiki/Poisson_distribution

which derives it as the limiting case of the binomial distribution. There is a simplification step which I am missing, which is the step(s) between

[tex]

\lim_{n\rightarrow\infty}\frac{n!}{k!(n-k)!}\left(\frac{\lambda}{n}\right)^k\left(1-\frac{\lambda}{n}\right)^{n-k}

[/tex]

and

[tex]

=\lim_{n\to\infty} \underbrace{\left({n \over n}\right)\left({n-1 \over n}\right)\left({n-2 \over n}\right) \cdots \left({n-k+1 \over n}\right)}\ \underbrace{\left({\lambda^k \over k!}\right)}\ \underbrace{\left(1-{\lambda \over n}\right)^n}\ \underbrace{\left(1-{\lambda \over n}\right)^{-k}}

[/tex]

Does the main simplification come from:

[tex]

\frac{n!}{k!(n-k)!} \left( \frac{\lambda}{n} \right)^k

[/tex]

[tex]

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

[/tex]

?

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# Simplifying Summation and Factorial

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