Solving Integrals by Series Expansions

[yfatehi]

In my PhD I need to solve an integral of the generalized MarcumQ function multiplied by a certain probability density function to get the overall event probability. Numerically solution produces bound result as it represents a probability but when trying to use a convergent power expansion of the MarcumQ function the result is divergent.

Is this logic?

 

[mathman]

Your statement is confusing. What is convergent and what is divergent (your statement sounds like you saying the same thing is both)?

You might supply some details.

 

[Mute]

In my PhD I need to solve an integral of the generalized MarcumQ function multiplied by a certain probability density function to get the overall event probability. Numerically solution produces bound result as it represents a probability but when trying to use a convergent power expansion of the MarcumQ function the result is divergent.

Is this logic?

You are not guaranteed to get a convergent series if you swap the order of an integral and an infinite sum, even if the original series was convergent. This could be what is happening, but it could also be that you may just need to double-check your work.

 

[yfatehi]

Form more clarity
I have a function F(a) expandable in power series. The parameter a is random variable and I must average by multiplying F(a) with the PDF of this variable and integrate from 0 to infinity. Although the PDF function is bound and the expansion of F(a) in series is convrgent, the resulting multiplication and integration is divergent series. This is my issue.
First is it mathematically possible or sure I made some mistake although i made the integration using mathematica

 

[mathman]

Is the series for F(a) uniformly convergent for all values of the parameter? If not that could be a problem.

 

[yfatehi]

How to test the uniform convergence

 

[mathman]

How to test the uniform convergence

If you use the basic δ ε method, then the choice has to be independent of the parameter.

 

[yfatehi]

If the initial series is uniform convergent in the region from 0 to inf. what is the status of the Expectation? ie E{f(x)}= Sum ( E{Un(x)} )

 

[mathman]

If the initial series is uniform convergent in the region from 0 to inf. what is the status of the Expectation? ie E{f(x)}= Sum ( E{Un(x)} )

What is the connection between f(x) and Un(x)? Also is x a random variable?

 

[yfatehi]

f(x)= Sum ( Un(x) ), n from 0 to infinity , x is random positive variable We need to get the average E{f(x)} over x so we first multiply with the Prob. Density function PDF(x) then integrate fom 0 to inf

first if the original series Un(x) is uniform convergent over x from 0 to inf, will its convergence be affected by mutiplication with PDF(x) of any shape?

Second if not will the integral after multiplication be convergent?

 

[chiro]

You are not guaranteed to get a convergent series if you swap the order of an integral and an infinite sum, even if the original series was convergent. This could be what is happening, but it could also be that you may just need to double-check your work.

To build up what Mute said, there is a condition that has to be met to do this.

Take a look at this page:

http://math.stackexchange.com/questions/48119/criteria-for-swapping-integration-and-summation-order