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#1 2013-06-19 00:31:48

ShivamS
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A horrid problem

Flip a coin 2N times, where N is large. Let P(x) be the probability of obtaining
exactly N + x heads. Show that P(x) = e^((-x^2)/N) divides by sqrt of pi times N


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#2 2013-06-19 01:11:38

bobbym
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Re: A horrid problem

Hi;

Is this what you want



to prove?


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#3 2013-06-19 01:52:44

anonimnystefy
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Re: A horrid problem

I don't think that would be correct, then.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#4 2013-06-19 02:04:07

bobbym
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Re: A horrid problem

I do not think so either.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#5 2013-06-19 02:12:58

anonimnystefy
Real Member

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Re: A horrid problem

It does seem to work without the minus sign, though.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#6 2013-06-19 02:18:19

Agnishom
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Re: A horrid problem

How do you do that?


'And fun? If maths is fun, then getting a tooth extraction is fun. A viral infection is fun. Rabies shots are fun.'
'God exists because Mathematics is consistent, and the devil exists because we cannot prove it'
'Who are you to judge everything?' -Alokananda

#7 2013-06-19 02:22:35

bobbym
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Re: A horrid problem

Hi;

Do what?

The answer is



that I can prove.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#8 2013-06-19 08:15:17

anonimnystefy
Real Member

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Re: A horrid problem

That is what the original problem is asking for. But I cannot get it. I am using the limit definition and Stirling's formula.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#9 2013-06-19 09:53:45

ShivamS
Super Member

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Re: A horrid problem

Post 7 is what I need proven/


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#10 2013-06-21 04:47:59

ShivamS
Super Member

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Re: A horrid problem

Can you prove it then?


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#11 2013-06-21 10:10:35

anonimnystefy
Real Member

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Re: A horrid problem

Hi Shivamcoder3013

Have you tried taking the limit as N goes to infinity of the ratio of the exact answer and the approximate one and proving it equals 1?


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#12 2013-06-21 11:48:02

bobbym
Administrator

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Re: A horrid problem

Hi;

The paper I am looking at "Gaussian and Coins."





Using Stirlings:








Notice the approximately equal sign that is because you are approximated a discrete distribution ( binomial ) with the Normal distribution.

1) is an approximation for 2) which the above steps prove. Even for large N it is still an approximation. When N approaches infinity 1) = 2).

To prove that you might need the limit but maybe since Stirlings formula is asymptotic to the factorial it might be implied in step 3.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#13 2013-06-22 02:19:52

ShivamS
Super Member

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Re: A horrid problem

I am not getting how you get that...


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#14 2013-06-22 03:44:15

bobbym
Administrator

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Re: A horrid problem

Hi Shivamcoder3013;

I am not getting much of the derivation either. It is a lot of algebra and undoubtedly was done with the help of a package. I put it down so you would have something.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

#15 2013-06-22 10:00:03

ShivamS
Super Member

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Re: A horrid problem

Ok, I will try to think about it a bit. Thanks a lot.


I have discovered a truly marvellous signature, which this margin is too narrow to contain. -Fermat
Give me a lever long enough and a fulcrum on which to place it, and I shall move the world. -Archimedes
Young man, in mathematics you don't understand things. You just get used to them. - Neumann

#16 2013-06-22 21:34:59

anonimnystefy
Real Member

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Re: A horrid problem

Hi bobbym

Have you tried getting the limit of the ratio of the two expressions (the exact one and the approximate one)? It does not approach 1.


The limit operator is just an excuse for doing something you know you can't.
“It's the subject that nobody knows anything about that we can all talk about!” ― Richard Feynman
“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

#17 2013-06-22 21:43:39

bobbym
Administrator

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Re: A horrid problem



yields 0.001079819330263761

The exact answer is:



yields 0.0010798643294


Seems pretty good. Try for larger n with x small in comparison to convince yourself numerically.

anonimnystefy wrote:

Have you tried getting the limit of the ratio of the two expressions (the exact one and the approximate one)? It does not approach 1.

I think the limit is 1.



According to M that is true. Why do you think the limit is not 1?

bobbym wrote:

To prove that you might need the limit but maybe since Stirlings formula is asymptotic to the factorial it might be implied in step 3.

Stirlings is an asymptotic form for the factorial. The limit of the ratio of Stirlings and the factorial is 1. The fact that he use Stirlings in his proof guarantees the above limit.


In mathematics, you don't understand things. You just get used to them.
I have the result, but I do not yet know how to get it.
All physicists, and a good many quite respectable mathematicians are contemptuous about proof.

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