- #1
rwinston
- 36
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I was looking at the web page containing a derivation for the Poisson distribution:
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]
?
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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