Guassian Distribution and How It Relates to Pi

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Guassian Distribution and How It Relates to Pi

#1  Postby Sityl » Aug 29, 2013 8:57 pm

I was reading the following, and didn't even get past the beggining since the example he used involves math which I haven't learned yet. I was wondering if anyone could, assuming no background in calculus or statistics, explain the relationship between gaussian distribution and pi.

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html wrote:THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."


A cursory google search of explanations seems to involve funky "f" from infinity to negative infinity, but I don't know what that means.

Thanks!
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Re: Guassian Distribution and How It Relates to Pi

#2  Postby scott1328 » Aug 29, 2013 9:17 pm

Although I wouldn't consider the HuffPost anything like authoritative on subjects of Math and Science, this article seems apropos to your question. Note that a Gaussian distribution is also called Normal distribution.

http://www.huffingtonpost.com/david-h-b ... 85725.html
The digits of pi, and of other well-known mathematical constants, have inspired mathematicians for centuries. Pi has been computed to prodigiously high precision, recently to more than 10 trillion digits by Alexander Yee and Shiguro Kondo, in part to facilitate research into such questions. It certainly appears that pi is a normal number, not only in base 2 and in base 10 but in other bases as well. But sadly, as noted above, there is no proof. (Some very recent empirical analysis of the first 16 trillion binary digits, corresponding to a bit less than 5 trillion decimal digits -- by us and several co-authors -- strongly suggests that pi is normal base 2, but again this is not a proof.)


The article links to this more formal PDF: http://www.carma.newcastle.edu.au/jon/normality.pdf
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Re: Guassian Distribution and How It Relates to Pi

#3  Postby campermon » Aug 29, 2013 10:27 pm

Sityl wrote:I was reading the following, and didn't even get past the beggining since the example he used involves math which I haven't learned yet. I was wondering if anyone could, assuming no background in calculus or statistics, explain the relationship between gaussian distribution and pi.

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html wrote:THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."


A cursory google search of explanations seems to involve funky "f" from infinity to negative infinity, but I don't know what that means.

Thanks!


I'll take your questions in reverse order and try to answer them in 'pub notation' :beer:

The funky 'f' is an elongated 'S' and basically means 'sum'. If you have a straight line graph and you want to know the area under the graph, you are basically working out the area of a triangle which is easy. If you have a curvy graph, it's a little trickier. What you do is break down the curve graph into a sequence of rectangles;

Image

Finding the area of rectangle is easy, so you do that and add (sum) them all up to find the total area.

Now then. If you have the maths expression for the curve, then you bugger about with some calculus to come up with an expression to calculate the area. In simple terms, the calculus bit slices the curve into an infinite number of infinitesimally thin rectangles that then add (sum) to make the area. Hence the big 's'.

Finally. For the gaussian normal distribution, we want to make the total area under the graph to equal 1. For example, the curve might represent the probability that a molecule might have a given speed. The molecule has to have some speed, so the probability that is has a speed somewhere on the graph is 1. The Pi comes in to the sums because it makes the area under the graph equal 1.

:cheers:
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Re: Guassian Distribution and How It Relates to Pi

#4  Postby Pulsar » Aug 29, 2013 10:29 pm

scott1328 wrote:Although I wouldn't consider the HuffPost anything like authoritative on subjects of Math and Science, this article seems apropos to your question. Note that a Gaussian distribution is also called Normal distribution.

That's an article about normal numbers, not about normal distributions. Those are two very different concepts.


The relation between the Gaussian distribution and π is a consequence of the fact that the total area under the curve exp(-x2) is equal to the square root of π. This is the standard proof:

http://en.wikipedia.org/wiki/Gaussian_integral#Careful_proof

The proof requires knowledge of double integration and polar coordinates. Sorry, I don't know how to explain it without using advanced calculus.
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Re: Guassian Distribution and How It Relates to Pi

#5  Postby scott1328 » Aug 29, 2013 10:33 pm

Pulsar wrote:
scott1328 wrote:Although I wouldn't consider the HuffPost anything like authoritative on subjects of Math and Science, this article seems apropos to your question. Note that a Gaussian distribution is also called Normal distribution.

That's an article about normal numbers, not about normal distributions. Those are two very different concepts.


The relation between the Gaussian distribution and π is a consequence of the fact that the total area under the curve exp(-x2) is equal to the square root of π. This is the standard proof:

http://en.wikipedia.org/wiki/Gaussian_integral#Careful_proof

The proof requires knowledge of double integration and polar coordinates. Sorry, I don't know how to explain it without using advanced calculus.

My apologies, I mistook your question, I have been seeing a meme floating around lately that assumes pi is a normal number and didn't take the time to understand what you were asking.
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Re: Guassian Distribution and How It Relates to Pi

#6  Postby lobawad » Aug 29, 2013 10:40 pm

campermon wrote:
Sityl wrote:I was reading the following, and didn't even get past the beggining since the example he used involves math which I haven't learned yet. I was wondering if anyone could, assuming no background in calculus or statistics, explain the relationship between gaussian distribution and pi.

http://www.dartmouth.edu/~matc/MathDrama/reading/Wigner.html wrote:THERE IS A story about two friends, who were classmates in high school, talking about their jobs. One of them became a statistician and was working on population trends. He showed a reprint to his former classmate. The reprint started, as usual, with the Gaussian distribution and the statistician explained to his former classmate the meaning of the symbols for the actual population, for the average population, and so on. His classmate was a bit incredulous and was not quite sure whether the statistician was pulling his leg. "How can you know that?" was his query. "And what is this symbol here?" "Oh," said the statistician, "this is pi." "What is that?" "The ratio of the circumference of the circle to its diameter." "Well, now you are pushing your joke too far," said the classmate, "surely the population has nothing to do with the circumference of the circle."


A cursory google search of explanations seems to involve funky "f" from infinity to negative infinity, but I don't know what that means.

Thanks!


I'll take your questions in reverse order and try to answer them in 'pub notation' :beer:

The funky 'f' is an elongated 'S' and basically means 'sum'. If you have a straight line graph and you want to know the area under the graph, you are basically working out the area of a triangle which is easy. If you have a curvy graph, it's a little trickier. What you do is break down the curve graph into a sequence of rectangles;

Image

Finding the area of rectangle is easy, so you do that and add (sum) them all up to find the total area.

Now then. If you have the maths expression for the curve, then you bugger about with some calculus to come up with an expression to calculate the area. In simple terms, the calculus bit slices the curve into an infinite number of infinitesimally thin rectangles that then add (sum) to make the area. Hence the big 's'.

Finally. For the gaussian normal distribution, we want to make the total area under the graph to equal 1. For example, the curve might represent the probability that a molecule might have a given speed. The molecule has to have some speed, so the probability that is has a speed somewhere on the graph is 1. The Pi comes in to the sums because it makes the area under the graph equal 1.

:cheers:


Ha! I actually understood that (but only because I can relate it digital sound synthesis, where I've picked up a little bit of maths by osmosis).

Thanks!
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Re: Guassian Distribution and How It Relates to Pi

#7  Postby campermon » Aug 29, 2013 10:45 pm

:beer:

The patented campermon beer mat physicsTM teaching method never fails.

:mrgreen:
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Re: Guassian Distribution and How It Relates to Pi

#8  Postby lobawad » Aug 29, 2013 10:46 pm

campermon wrote::beer:

The patented campermon beer mat physicsTM teaching method never fails.

:mrgreen:


:cheers:

Sierra Nevada Pale Ale here!
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Re: Guassian Distribution and How It Relates to Pi

#9  Postby campermon » Aug 29, 2013 10:49 pm

:beer:

Nothing special here, just Fosters lager (had a chicken curry earlier on :hungry: ). Should have got some Cobra in! :doh:
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Re: Guassian Distribution and How It Relates to Pi

#10  Postby tuco » Aug 29, 2013 11:27 pm

Pi is just a myth:



:p
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Re: Guassian Distribution and How It Relates to Pi

#11  Postby someguy1 » Aug 30, 2013 9:21 am

Sityl wrote:
A cursory google search of explanations seems to involve funky "f" from infinity to negative infinity, but I don't know what that means.



Others have said this but perhaps not as explicitly.

You know the "bell curve" that always turns up when you're graphing the distribution of some characteristic of a population? Very few (adult) people are 4 feet tall and very few are 7 feet tall. The vast majority are in the middle.

Same for the number of people with a given IQ -- there are some dummies and some geniuses but the rest of us are somewhere in the middle.

Well, if you take the area under the bell curve, that area turns out to be the square root of pi.

Yeah I know, that's crazy! That's what the anecdote is talking about.

The funky S is the integral sign. It's just how mathematicians figure out the area under a curve. And the negative and positive infinity symbols just mean that they're computing the area all along the entire x-axis, from negative infinity to positive infinity.
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Re: Guassian Distribution and How It Relates to Pi

#12  Postby susu.exp » Aug 31, 2013 7:40 am

lobawad wrote:Ha! I actually understood that (but only because I can relate it digital sound synthesis, where I've picked up a little bit of maths by osmosis).

Thanks!


In that case you might like the following: You can do a fourier transformation of any function. The normal distribution has the property that it is transformed into a normal distribution with inverse variance. I.e. a narrow distribution is transformed into a very broad one. This is used in the proof of the central limit theorem - instead of directly showing that a sum of centered iid random variables divided by the square root of their number converges in distribution to a normal distribution you show that the fourier transform of that normalized sum converges in distribution to a normal distribution.
It´s also at the heart of accoustic uncertaincy - the fourier transformation of an amplitude vs time is of course a graph showing amplitude vs. frequency. And if a normal distribution in one turns into a normal distribution with inverse variance in the other then the more exact you can pinpoint time, the less exact you can pinpoit frequency and vice versa. That of course turns up in quantum mechanics as well - time and energy, momentum and position... If there´s a formula that says the product of the errors for two values is greater or equal than a constant, this property of the normal distribution is at the heart of it.

To get to the OP a bit: The gaussian distribution uses an ex function. And one important thing to note here is that eix=cos(x)+i sin(x), i.e. for real x, it produces a unit circle in the complex plane! So if your problem involved e, pi lurks right around the corner.
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Re: Guassian Distribution and How It Relates to Pi

#13  Postby Sityl » Aug 31, 2013 8:17 am

Thanks! :D
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Re: Guassian Distribution and How It Relates to Pi

#14  Postby lobawad » Aug 31, 2013 12:05 pm

susu.exp wrote:
lobawad wrote:Ha! I actually understood that (but only because I can relate it digital sound synthesis, where I've picked up a little bit of maths by osmosis).

Thanks!


In that case you might like the following: You can do a fourier transformation of any function. The normal distribution has the property that it is transformed into a normal distribution with inverse variance. I.e. a narrow distribution is transformed into a very broad one. This is used in the proof of the central limit theorem - instead of directly showing that a sum of centered iid random variables divided by the square root of their number converges in distribution to a normal distribution you show that the fourier transform of that normalized sum converges in distribution to a normal distribution.
It´s also at the heart of accoustic uncertaincy - the fourier transformation of an amplitude vs time is of course a graph showing amplitude vs. frequency. And if a normal distribution in one turns into a normal distribution with inverse variance in the other then the more exact you can pinpoint time, the less exact you can pinpoit frequency and vice versa. That of course turns up in quantum mechanics as well - time and energy, momentum and position... If there´s a formula that says the product of the errors for two values is greater or equal than a constant, this property of the normal distribution is at the heart of it.

To get to the OP a bit: The gaussian distribution uses an ex function. And one important thing to note here is that eix=cos(x)+i sin(x), i.e. for real x, it produces a unit circle in the complex plane! So if your problem involved e, pi lurks right around the corner.


That's a very nice and clean explanation of a classic dilemma in audio analysis (for purposes of resynthesis, for example), thanks.

For the musician it means that either you're either going to get a good analysis of amplitude variation over time ("tight envelope follower") OR a good analysis of pitch ("pitch tracker"). A single analysis will always give you a compromise.

For my usual purposes, "playing" a synthesizer with an acoustic instrument (i.e., parameters for synthesis derived from acoustic performance) the solution is obvious, but computationally expensive. I just run one analysis of the acoustic performance geared toward amplitude and another for pitch.

I could go on at great length about all the interesting problems and solutions here, but I think that this is not the place.
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