0

I would appreciate if you could help me to find the following integral:

$$f(u)= \int_{-\infty }^{\infty} {{\rm e}^{{\dfrac {t \left( -t{\mu_{{x}}}^{2}{\sigma_{{y}}}^{2}-t{ \mu_{{y}}}^{2}{\sigma_{{x}}}^{2}+2\,i\mu_{{x}}\mu_{{y}} \right) }{2({t}^ {2}{\sigma_{{x}}}^{2}{\sigma_{{y}}}^{2}+1)}}}}{\frac {e^{-itu}}{\sqrt {{t}^{2} {\sigma_{{x}}}^{2}{\sigma_{{y}}}^{2}+1}}} \;dt$$

May
  • 1,254
  • 1
    This question looks like somebody very troubled's worst nightmare come true, but if you post such a question without giving somebackground ,explanation, showing some effort orself work I greatly doubt many will give it a serious thought... – DonAntonio Sep 17 '13 at 19:32
  • @DonAntonio you are right! It's been a few days now that I've got stuck in finding this integral! Actually my question is how to find the PDF of product of two normal random variables. Knowing its characteristic function (http://math.stackexchange.com/questions/496465/characteristic-function-of-random-variable-z-xy-where-x-and-y-are-independent) ,I want to find its PDF. – May Sep 17 '13 at 19:48
  • and PDF is the fourier transform of the characteristic function. – May Sep 17 '13 at 20:19

1 Answers1

0

The integral could be evaluated in closed form if one could evaluate $$\int_0^{\infty}\cos(a u)e^{-bu/\sqrt{u^2+1}}du$$. This is doubtful.

larry
  • 834