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#1 2013-06-18 02:31:48

ShivamS
Member
Registered: 2011-02-07
Posts: 3,528

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

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#2 2013-06-18 03:11:38

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#3 2013-06-18 03:52:44

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,518

Re: A horrid problem

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


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#4 2013-06-18 04:04:07

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

Re: A horrid problem

I do not think so either.


In mathematics, you don't understand things. You just get used to them.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#5 2013-06-18 04:12:58

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,518

Re: A horrid problem

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


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#6 2013-06-18 04:18:19

Agnishom
Real Member
From: The Complex Plane
Registered: 2011-01-29
Posts: 16,574
Website

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'
'Humanity is still kept intact. It remains within.' -Alokananda

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#7 2013-06-18 04:22:35

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#8 2013-06-18 10:15:17

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,518

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.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#9 2013-06-18 11:53:45

ShivamS
Member
Registered: 2011-02-07
Posts: 3,528

Re: A horrid problem

Post 7 is what I need proven/

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#10 2013-06-20 06:47:59

ShivamS
Member
Registered: 2011-02-07
Posts: 3,528

Re: A horrid problem

Can you prove it then?

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#11 2013-06-20 12:10:35

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,518

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?


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#12 2013-06-20 13:48:02

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#13 2013-06-21 04:19:52

ShivamS
Member
Registered: 2011-02-07
Posts: 3,528

Re: A horrid problem

I am not getting how you get that...

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#14 2013-06-21 05:44:15

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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#15 2013-06-21 12:00:03

ShivamS
Member
Registered: 2011-02-07
Posts: 3,528

Re: A horrid problem

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

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#16 2013-06-21 23:34:59

anonimnystefy
Real Member
From: The Foundation
Registered: 2011-05-23
Posts: 15,518

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.


“Here lies the reader who will never open this book. He is forever dead.

“Taking a new step, uttering a new word, is what people fear most.” ― Fyodor Dostoyevsky, Crime and Punishment

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#17 2013-06-21 23:43:39

bobbym
Administrator
From: Bumpkinland
Registered: 2009-04-12
Posts: 86,161

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.
Of course that result can be rigorously obtained, but who cares?
Combinatorics is Algebra and Algebra is Combinatorics.

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