Math Is Fun Forum

ac

Member

Registered: 2009-08-19

Posts: 8

another tricky integral

Dear all,

I'm trying to solve this integral:

with

constants and real

Please, help me!

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: another tricky integral

Hi ac;

Are you working with some probability density function? Did you start with a definite integral?

In order to have a chance to integrate that you have to work on this expression

You would need to eliminate u2 and the s2, is that possible?

Last edited by bobbym (2009-08-20 05:35:52)

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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.

George,Y

Member

Registered: 2006-03-12

Posts: 1,379

Re: another tricky integral

orz, I don't even know what it is for in probability

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X'(y-Xβ)=0

ac

Member

Registered: 2009-08-19

Posts: 8

Re: another tricky integral

Hi Bobbym,

Thank you for your answer.

You can look to the function and if it is normalized, you have a probability density function... correct.

About the Gaussian, all the fun is in u2 and s2, without them you don't have the square root of a sum, the other part of the fun...

Is it impossible?

ac

Member

Registered: 2009-08-19

Posts: 8

Re: another tricky integral

Hi George,

Thank you for your interest.

Do I need to have a use for it?
I'm trying to solve this integral just as an exercise.

Is it impossible?

Last edited by ac (2009-08-28 07:49:32)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: another tricky integral

Hi ac;

I would say that it's not possible to get a closed form for it. When it is normalized, as you know the integral will be equal to 1. Also it will now be a definite integral as I suspected. It will be possible using the methods of numerical integration and asymptotic analysis to get the area for any a,b.

---

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.

ac

Member

Registered: 2009-08-19

Posts: 8

Re: another tricky integral

Hi Bobbym,

Thank you!

Numerical integration and asymptotic analysis.

If

have multiple dimensions...
Well I have to go to a casino in Monte Carlo...

or visit Laurent series and 

function...
asymptotic analysis, I wasn't looking in that direction... THANK YOU!

Can I ask for a good book in that subject?

Last edited by ac (2009-08-28 20:31:11)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: another tricky integral

Hi ac;

You don't have to worry about monte-carlo integration because you don't have a multiple dimensions. You only a single integral. Good old gaussian integration is the way to go but there are tons of other methods, simpsons, trapezoidal, rombergs are just a few of dozens.

Books, any numerical analsis text that you can understand will cover all the forms of numerical integration.

The principal part (negative portion of a laurent series) will provide asymptotic forms but their are other methods such as IBP, bootstrapping and many more. Sadly, I cannot recommend any really good books on asymptotic analysis. Although I have gone through several, they are not meant for someone to teach themselves with. That is how I classify a good book, whether or not you can learn the subject by reading it yourself. You will have to learn it a little piece at a time, from many books, lectures, and articles on the net.

Luckily, you will not need it for this problem, gaussian integration and a computer will get the job done. Of course I am talking about a definite integral which a PDF or CDF is.

Last edited by bobbym (2009-08-28 23:04:56)

---

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.

ac

Member

Registered: 2009-08-19

Posts: 8

Re: another tricky integral

Hi Bobbym,

You are right, the function is 1D, but it is easy to increase dimensions...

My goal is not to "get the job done", but to get the job done in a smart way... the goal is a closed form...

Do you think IBP can help in this case? any suggestion?

Thanks

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: another tricky integral

Hi ac;

You are right, the function is 1D, but it is easy to increase dimensions...

I can't say that is true. You would need

to have a 2 dimensional integral and

for a 3D one etc. Your function is just of x.

My goal is not to "get the job done", but to get the job done in a smart way... the goal is a closed form...

Yes a closed form would be nice but it might not possible. Most integrals do not have closed forms. I have already tried all the methods that I am aware of. Also there is no reason to think that an analytical solution is smarter than a numerical one. What if the closed form is hundreds of pages long. What if the closed form contains functions that are difficult to compute.

Something might be possible if you tell me what the limits of integration are. Then I can think of approximating that integrand with a simpler function. Or maybe computing a table of values and curve fitting those. Otherwise unless I know the limits of integration I cannot fit anything to that integrand. Probablity density functions typically go from -∞ to ∞. Is that what the original problems limits of integration were?

Last edited by bobbym (2009-08-29 06:48:25)

---

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.

ac

Member

Registered: 2009-08-19

Posts: 8

Re: another tricky integral

Hi Bobbym,

Thank you, for your answer.

it is just the notation...

And yes, the limits of integration are from -∞ to ∞.

Thanks!

Last edited by ac (2009-08-30 22:29:29)

bobbym

bumpkin

From: Bumpkinland

Registered: 2009-04-12

Posts: 109,606

Re: another tricky integral

Hi ac;

You lost me there.

And yes, the limits of integration are from -∞ to ∞.

So, we have a definite integral, which in this case is nothing but an area. We are in position to do some numerical work. Provide me with the normalizing constants so I don;t have to compute them. Tell me more about the particulars of the problem so that I can determine the best way to approximate the integrand.

---

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.