My Math Forum Gaussian integral

 October 17th, 2013, 11:24 AM #1 Newbie   Joined: Oct 2013 Posts: 3 Thanks: 0 Gaussian integral Can anyone help with the following integral? $\int_0^{\infty} x^2 \phi (ax + b)\Phi (cx + d)dx$ Thanks.
 October 17th, 2013, 12:11 PM #2 Global Moderator   Joined: May 2007 Posts: 6,770 Thanks: 700 Re: Gaussian integral You need to define ? and ?.
 October 17th, 2013, 12:49 PM #3 Newbie   Joined: Oct 2013 Posts: 3 Thanks: 0 Re: Gaussian integral The functions in the integrand are PDF and CDF of a standard normal distribution, respectively.
 October 18th, 2013, 12:23 PM #4 Global Moderator   Joined: May 2007 Posts: 6,770 Thanks: 700 Re: Gaussian integral My gut feeling - there is no closed form solution.
 October 18th, 2013, 07:41 PM #5 Newbie   Joined: Oct 2013 Posts: 3 Thanks: 0 Re: Gaussian integral I calculated the integral with several problems numerically and obtained solutions. I think a closed form solution may exist. Other similar integrals can be found at http://en.wikipedia.org/wiki/List_of_in ... _functions.
 October 19th, 2013, 12:44 PM #6 Global Moderator   Joined: May 2007 Posts: 6,770 Thanks: 700 Re: Gaussian integral Looking at the reference, it looks like you have two difficulties to overcome. You have an x^2 term (although this probably could be handled) and the limits are (0,?).

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