Question

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.

Answer 1

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.