Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

I'm trying to find the first moment of the following function: $f(x) = \frac{(-ax+\sqrt{1-a^2})(-bx+\sqrt{1-b^2})}{\sqrt{x^2+1}}H(-ax+\sqrt{1-a^2})H(-bx+\sqrt{1-b^2})$ where $H(x)$ denotes the Heaviside function, and having $a,b \in ]-1,1[$ and $x \sim \mathcal{N}(\mu, \sigma)$. Notice that the heavisides simply nullify the expression whenever one of the two terms in the numerator is less than 0. So far I have been unable to find the characteristic function or an antiderivative for this expression with the pdf (I tried to express the numerator as a linear combination of Hermite polynomials). Since I'm out of ideas right now, I am asking for possible help/advice and/or some literature dealing with this kind of problem.

share|improve this question
add comment

1 Answer 1

up vote 0 down vote accepted

The best way to proceed is to use Legendre polynomials through the formula $$ \frac{1}{\sqrt{1-2yx+x^2}} = \sum_{n=0}^\infty P_n(y) x^n. $$ In this way you will get closed form integrals with the error function.

share|improve this answer
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.