Monte Carlo Integration: Error Estimate Reliability for Non-Square Integrables

[trelek2]

Hi!

I need help with the monte carlo integration: reliability of the error estimate for functions that are not square integrable.

I'm supposed to investigate this topic.*Hence my first question is what is a function that is not square integrable? I found that such a function is 1/sqrt(x) on the interval 0 to 1. Apparently a function is not square integrable if the integral of its absolute value squared is not finite on that integral... I thought the for f(x)= 1/sqrt(x) that will be -1?
Anyway I evaluated the integrals for 1/sqrt(x) from 0 to 1 (which is 2 analytically) for dofferent number of sample points. Indeed the estimated errors are nowhere close the actual errors...

Can anyone explain why does this happen? And why is 1/sqrt(x) not square integrable?

 

[mathman]

Hi!

I need help with the monte carlo integration: reliability of the error estimate for functions that are not square integrable.

I'm supposed to investigate this topic.*Hence my first question is what is a function that is not square integrable? I found that such a function is 1/sqrt(x) on the interval 0 to 1. Apparently a function is not square integrable if the integral of its absolute value squared is not finite on that integral... I thought the for f(x)= 1/sqrt(x) that will be -1?
Anyway I evaluated the integrals for 1/sqrt(x) from 0 to 1 (which is 2 analytically) for dofferent number of sample points. Indeed the estimated errors are nowhere close the actual errors...

Can anyone explain why does this happen? And why is 1/sqrt(x) not square integrable?

The integral of 1/x is infinite. Therefore the variance of your estimator is infinite. That is why Monte Carlo doesn't work.

 

[trelek2]

Will that be also true for exp(-x^2)? I get a NaN as the error estimate oddly only in this case.

 

[mathman]

Will that be also true for exp(-x^2)? I get a NaN as the error estimate oddly only in this case.

Several questions:

1. What is the domain of x?
2. Exactly what is the prob. distribution function (or prob. density function)?
3. What is NaN (Sodium Nitride?)?

 

[ideasrule]

In computing, NaN stands for "not a number", which usually means an infinity or undefined value has popped up.

 

[mathman]

If exp(-x^2) is supposed to be your density function (although there will be a constant attached to it to make it integrate to 1), then the variance is known.
exp(-.5x^2)/ √(2π) is the density for the standard normal with mean 0 and variance 1.