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From: 2.7182818284590... on 17 Feb 2010 12:38
I noticed that when I plot the numbers on a given row of Pascal's
Triangle, the curve generated resembles the Bell Curve.

However, what distinguishes the Pascal's Triangle to the Normal
Gaussian Distribution Curve is this: When I divide the biggest number
(which is in the center of Pascal's Triangle) by the SUM of the entire
row, this value seems to approach 0 the further down on the pyramid
that I go. On the other hand, The Gaussian Distribution has a value
of ~0.4 at x=0.

If the two are related, please explain how.
From: Jan Kristian Haugland on 17 Feb 2010 12:53
On 17 Feb, 18:38, "2.7182818284590..." <tangent1...(a)gmail.com> wrote:
> I noticed that when I plot the numbers on a given row of Pascal's
> Triangle, the curve generated resembles the Bell Curve.
>
> However, what distinguishes the Pascal's Triangle to the Normal
> Gaussian Distribution Curve is this:  When I divide the biggest number
> (which is in the center of Pascal's Triangle) by the SUM of the entire
> row, this value seems to approach 0 the further down on the pyramid
> that I go.  On the other hand, The Gaussian Distribution has a value
> of ~0.4 at x=0.
>
> If the two are related, please explain how.

(nCk) / 2^n is approximately exp(-2((k-n/2)^2)/n) / sqrt(pi*n/2)

---
J K Haugland
http://www.neutreeko.net

From: Frederick Williams on 17 Feb 2010 13:13
"2.7182818284590..." wrote:
>
> I noticed that when I plot the numbers on a given row of Pascal's
> Triangle, the curve generated resembles the Bell Curve.
>
> However, what distinguishes the Pascal's Triangle to the Normal
> Gaussian Distribution Curve is this: When I divide the biggest number
> (which is in the center of Pascal's Triangle) by the SUM of the entire
> row, this value seems to approach 0 the further down on the pyramid
> that I go. On the other hand, The Gaussian Distribution has a value
> of ~0.4 at x=0.
>
> If the two are related, please explain how.

Perhaps you have discovered the normal approximation to the binomial
distribution. It is a special case of the central limit theorem known
as the De Moivre-Laplace limit theorem.

There are tables and diagrams in Feller, vol I.

--
.... A lamprophyre containing small phenocrysts of olivine and
augite, and usually also biotite or an amphibole, in a glassy
groundmass containing analcime.
From: tadchem on 17 Feb 2010 20:05
On Feb 17, 12:38 pm, "2.7182818284590..." <tangent1...(a)gmail.com>
wrote:
> I noticed that when I plot the numbers on a given row of Pascal's
> Triangle, the curve generated resembles the Bell Curve.
>
> However, what distinguishes the Pascal's Triangle to the Normal
> Gaussian Distribution Curve is this:  When I divide the biggest number
> (which is in the center of Pascal's Triangle) by the SUM of the entire
> row, this value seems to approach 0 the further down on the pyramid
> that I go.  On the other hand, The Gaussian Distribution has a value
> of ~0.4 at x=0.
>
> If the two are related, please explain how.

Google "binomial distribution," "discrete distribution," "continuous
distribution," "Gaussian distribution," and then the names of all the
other distributions you stumble across - uniform, geometric,
multinomial, Poisson, hypergeometric, Pascal, gamma, exponential,
beta, chi-squared, Student's, and Snedecor's should keep you busy for
a while.

Tom Davidson
Richmond, VA
From: mike3 on 17 Feb 2010 20:14
On Feb 17, 6:05 pm, tadchem <tadc...(a)comcast.net> wrote:
> On Feb 17, 12:38 pm, "2.7182818284590..." <tangent1...(a)gmail.com>
> wrote:
>
> > I noticed that when I plot the numbers on a given row of Pascal's
> > Triangle, the curve generated resembles the Bell Curve.
>
> > However, what distinguishes the Pascal's Triangle to the Normal
> > Gaussian Distribution Curve is this:  When I divide the biggest number
> > (which is in the center of Pascal's Triangle) by the SUM of the entire
> > row, this value seems to approach 0 the further down on the pyramid
> > that I go.  On the other hand, The Gaussian Distribution has a value
> > of ~0.4 at x=0.
>
> > If the two are related, please explain how.
>
> Google "binomial distribution," "discrete distribution," "continuous
> distribution," "Gaussian distribution," and then the names of all the
> other distributions you stumble across - uniform, geometric,
> multinomial, Poisson, hypergeometric, Pascal, gamma, exponential,
> beta, chi-squared, Student's, and Snedecor's should keep you busy for
> a while.
>

So all those things are needed to put together the complete theory of
the relationship?
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