can any one help me with following integral? I need closed-form

\begin{equation} \int_{-\infty}^{\infty} \frac{(\phi(x))^2}{\Phi(x)} \phi(ax+b) dx \end{equation}

where $\phi(x)= \frac{1}{\sqrt{(2 \pi)}} e^{-x^2/2}$ is the pdf of a standard normal distribution and $\Phi(x)= \int_{-\infty}^x \phi(t)dt$ is the CDF of a standard normal distribution.

  • $\begingroup$ That $\frac{1}{\Phi(x)}$ term in the integrand function is highly suspicious. Which probability are you trying to compute? My bet is that $\Phi(x)$ is in the wrong place. $\endgroup$ – Jack D'Aurizio Nov 13 '17 at 13:05
  • $\begingroup$ @JackD'Aurizio no sir, it is in right place. I am sure. I am calculating a Fisher information. $\endgroup$ – Alireza Nov 14 '17 at 4:40

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