Math Is Fun Forum

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

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

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: A horrid problem

Hi;

Is this what you want

to prove?

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In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

Re: A horrid problem

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

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“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
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: A horrid problem

I do not think so either.

---

In mathematics, you don't understand things. You just get used to them.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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
The knowledge of some things as a function of age is a delta function.

Agnishom

Real Member

From: Riemann Sphere

Registered: 2011-01-29

Posts: 25,002

Website

Re: A horrid problem

How do you do that?

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'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'
I'm not crazy, my mother had me tested.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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
The knowledge of some things as a function of age is a delta function.

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: A horrid problem

Post 7 is what I need proven/

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: A horrid problem

Can you prove it then?

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: A horrid problem

I am not getting how you get that...

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.

ShivamS

Member

Registered: 2011-02-07

Posts: 3,648

Re: A horrid problem

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

anonimnystefy

Real Member

From: Harlan's World

Registered: 2011-05-23

Posts: 16,049

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
The knowledge of some things as a function of age is a delta function.

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

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.

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?

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.
If it ain't broke, fix it until it is.
Always satisfy the Prime Directive of getting the right answer above all else.