I read this in a book (it was stats and about poisson approx to normal)(adsbygoogle = window.adsbygoogle || []).push({});

Given was this:

[tex]n(n-1)(n-2) \cdots (n-r+1) = \frac{n!}{(n-r)!} \approx n^r[/tex]

Stating that "Stirling's approximation" had been used.

So I looked the up and found:

[tex]\ln n! \approx n\ln n - n\ [/tex]

In the poisson distribution n is very large and [tex]r[/tex] is very small compared to [tex]n[/tex] so all the terms in the given equation approximate to [tex]n[/tex]... This gives me my [tex]\approx n^r[/tex]

But I just wondered where the Stirling equation comes in to it...

[tex]\ln (\frac{n!}{(n-r)!}) = \ln(n!) - \ln((n-r)!) \ [/tex]

[tex]\approx n\ln n - n - \left[ (n-r)\ln((n-r)) - (n-r) \right]\ [/tex]

[tex]\approx n\ln n - n - (n-r)\ln((n-r)) + n - r \ [/tex]

[tex]\approx n\ln n - (n-r)\ln((n-r)) -r \ [/tex]

...

That's as far as I got...

[tex]\approx \ln (n^n) - \ln((n-r)^{(r-n)}) -r \ [/tex]

Unless taking logs, instead of to base e, to base n...

[tex]\approx Log_n (n^n) - Log_n ((n-r)^{(r-n)}) -r \ [/tex]

Then...

[tex]Log_n (n^n) - Log_n ((n-r)^{(r-n)}) = r\ [/tex]

[tex]n^n - (n-r)^{(n-r)} = n^r\ [/tex]

^ not sure if that's correct though

Can anyone help?

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Stirling's approximation proof?

Loading...

Similar Threads for Stirling's approximation proof |
---|

A Is the proof of these results correct? |

I Doubt about proof on self-adjoint operators. |

I Addition of exponents proof in group theory |

B Help understanding a proof |

**Physics Forums | Science Articles, Homework Help, Discussion**