Deriving the Bell Curve

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Begin with a row from the Pascal triangle, preferably some large exponent , derive the Gaussian Distribution.

Solution

The bell curve is a probability density curve of binary systems. Then the probability at a some displacement from the medium is P(n,k)=(nk)2n=n!(12n+k)!(12nk)!2nP(n, k) = \left( \begin{matrix} n \\ k \end{matrix} \right) {2}^{-n}= \frac{n!}{(\frac{1}{2}n + k)! (\frac{1}{2}n - k)! {2}^{n}}

Using the Stirling approximation and treating k=σ2k = \frac{\sigma}{2}, we have
P(n,σ)(n2π)12(n2)n(n2σ24)12(n+1)(n+σnσ)σ2.P(n, \sigma) \sim {\left(\frac{n}{2\pi} \right)}^{\frac{1}{2}} {\left(\frac{n}{2}\right)}^{n} {\left(\frac{{n}^{2} - {\sigma}^{2}}{4}\right)}^{-\frac{1}{2}(n+1)}{\left(\frac{n + \sigma}{n-\sigma}\right)}^{\frac{-\sigma}{2}}.

For n>>σn>>\sigma , n+σnσ1+2σn\frac{n + \sigma}{n-\sigma} \sim 1+\frac{2\sigma}{n}; hence, for large nn P(n,σ)(n2π)12(1σ2n2)12(n+1)(1+2σn)σ2.P(n, \sigma) \sim {\left(\frac{n}{2\pi} \right)}^{\frac{1}{2}} {\left(1- \frac{{\sigma}^{2}}{{n}^{2}}\right)}^{-\frac{1}{2}(n+1)}{\left(1+\frac{2\sigma}{n}\right)}^{\frac{-\sigma}{2}}.

Taking the logarithm yields ln(P(n,σ))12ln(2πn)12(n+1)ln(1σ2n2)σ2ln(1+2σn).ln(P(n,\sigma)) \sim \frac{1}{2}ln \left (\frac{2}{\pi n}\right) - \frac{1}{2}(n+1)ln \left (1- \frac{{\sigma}^{2}}{{n}^{2}}\right) - \frac{\sigma}{2}ln \left (1+\frac{2\sigma}{n}\right).

For small xx, ln(1+x)xln(1+x) \approx x; subsequently, ln(P(n,σ))12ln(2πn)12(n+1)(σ2n2)σ2(2σn)ln(P(n,\sigma)) \sim \frac{1}{2}ln \left (\frac{2}{\pi n}\right) - \frac{1}{2}(n+1) \left (-\frac{{\sigma}^{2}}{{n}^{2}}\right) - \frac{\sigma}{2} \left (\frac{2\sigma}{n}\right) or ln(P(n,σ))12ln(2πn)+σ2n2σ22n.ln(P(n,\sigma)) \sim \frac{1}{2}ln \left (\frac{2}{\pi n}\right) + \frac{{\sigma}^{2}}{{n}^{2}} - \frac{{\sigma}^{2}}{2n}.

Since σ2n2\frac{{\sigma}^{2}}{{n}^{2}} vanishes faster than σ22n\frac{{\sigma}^{2}}{2n} for very large nn, we arrive at the result:

P(n,σ)=(2πn)12eσ22n.P(n, \sigma) = {\left(\frac{2}{\pi n} \right)}^{\frac{1}{2}} {e}^{\frac{-{\sigma}^{2}}{2n}}.

Check out my other notes at Proof, Disproof, and Derivation

Note by Steven Zheng
5 years, 3 months ago

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