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From: 2.7182818284590... on 9 Mar 2010 17:11
What is the relationship between the Gaussian Curve and the
coefficients of Pascal's Triangle? When I plot the coefficients of PT
for the 15th row, it looks very much like a Gaussian Distribution.



1 16
16 15
120 14
560 13
1820 12
4368 11
8008 10
11440 9
12870 8
11440 7
8008 6
4368 5
1820 4
560 3
120 2
16 1
1 0


So how are these two phenomenon's related? Coincidentally, or is the
Binomial Distribution the basis for the Gaussian Distribution?
From: Ray Vickson on 9 Mar 2010 20:10
On Mar 9, 2:11 pm, "2.7182818284590..." <tangent1...(a)gmail.com> wrote:
> What is the relationship between the Gaussian Curve and the
> coefficients of Pascal's Triangle?  When I plot the coefficients of PT
> for the 15th row, it looks very much like a Gaussian Distribution.
>
> 1       16
> 16      15
> 120     14
> 560     13
> 1820    12
> 4368    11
> 8008    10
> 11440   9
> 12870   8
> 11440   7
> 8008    6
> 4368    5
> 1820    4
> 560     3
> 120     2
> 16      1
> 1       0
>
> So how are these two phenomenon's related?  Coincidentally, or is the
> Binomial Distribution the basis for the Gaussian Distribution?

Congratulations. You have noticed something that was known to De
Moivre back in 1738; see the De Moivre Laplace Limit Theorem,
http://en.wikipedia.org/wiki/De_Moivre%E2%80%93Laplace_theorem or
http://mathworld.wolfram.com/deMoivre-LaplaceTheorem.html .

R.G. Vickson
From: Frisbieinstein on 9 Mar 2010 20:55
On Mar 10, 6:11 am, "2.7182818284590..." <tangent1...(a)gmail.com>
wrote:
> What is the relationship between the Gaussian Curve and the
> coefficients of Pascal's Triangle?  When I plot the coefficients of PT
> for the 15th row, it looks very much like a Gaussian Distribution.
>
> 1       16
> 16      15
> 120     14
> 560     13
> 1820    12
> 4368    11
> 8008    10
> 11440   9
> 12870   8
> 11440   7
> 8008    6
> 4368    5
> 1820    4
> 560     3
> 120     2
> 16      1
> 1       0
>
> So how are these two phenomenon's related?  Coincidentally, or is the
> Binomial Distribution the basis for the Gaussian Distribution?

They are very similar. You can think of the Gaussian as the limit as
the number of trials goes to infinity.
From: 2.7182818284590... on 10 Mar 2010 12:51
On Mar 9, 8:55 pm, Frisbieinstein <patmpow...(a)gmail.com> wrote:
> On Mar 10, 6:11 am, "2.7182818284590..." <tangent1...(a)gmail.com>
> wrote:
>
>
>
> > What is the relationship between the Gaussian Curve and the
> > coefficients of Pascal's Triangle?  When I plot the coefficients of PT
> > for the 15th row, it looks very much like a Gaussian Distribution.
>
> > 1       16
> > 16      15
> > 120     14
> > 560     13
> > 1820    12
> > 4368    11
> > 8008    10
> > 11440   9
> > 12870   8
> > 11440   7
> > 8008    6
> > 4368    5
> > 1820    4
> > 560     3
> > 120     2
> > 16      1
> > 1       0
>
> > So how are these two phenomenon's related?  Coincidentally, or is the
> > Binomial Distribution the basis for the Gaussian Distribution?
>
> They are very similar. You can think of the Gaussian as the limit as
> the number of trials goes to infinity.

Very good!!! I had that feeling as well. Do you know where I can
find the proof of this (i.e. Gaussian Distribution is a Binomial
Distribution with infinity outcomes)?
From: Frederick Williams on 10 Mar 2010 13:56
"2.7182818284590..." wrote:

>
> Very good!!! I had that feeling as well. Do you know where I can
> find the proof of [the deMoivre-Laplace limit theorem]?

Feller, An Introduction to Probability Theory and Its Application,
volume I, chapter VII, section 3.

--
I can't go on, I'll go on.
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