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Is there an analytic solution for the following integral?

$$ I = \int_{-\infty}^{\infty} x^2 \left\{ \Phi (a + b x) \right\}^2 \phi(x) \mathrm{d}x,$$

where $\Phi(\cdot)$ and $\phi(\cdot)$ are the cdf and pdf respectively of the standard normal.

Wikipedia has a list of related integrals (https://en.wikipedia.org/wiki/List_of_integrals_of_Gaussian_functions) but not the one I want to compute.

Does anyone have a way to solve this integral analytically? Thanks!

gigo
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    Welcome to MSE! Here we appreciate context and your own thoughts or attempts. Unfortunately this question will get closed otherwise, so to prevent this, please include more information into your question body. – vitamin d May 07 '21 at 12:17
  • By $\Phi(a+bx)^2$ did you really mean $\Phi((a+bx)^2)$? Because otherwise this integral certainly won't have a closed form. – K.defaoite May 07 '21 at 12:51
  • $ \Phi (a + b x)^2$ means $ \left{ \Phi (a + b x) \right} ^2$. I modified the formula to make it easier to understand. – gigo May 07 '21 at 13:08
  • Does ${.}$ mean the fractional part of the function or are they just parentheses? – MasB May 07 '21 at 13:11
  • Just parentheses. – gigo May 07 '21 at 13:16
  • It would help if you wrote out what the functions $\phi(x)$ and $\Phi(x)$ are – Sal May 07 '21 at 14:21
  • $ \Phi(x) $ and $ \phi(x) $ are the cdf and pdf respectively of the standard normal. $ \phi(x) = \frac{1}{\sqrt{2\pi}} \exp(-\frac{x^2}{2}) $, $\Phi(x) = \int_{-\infty}^x \phi(t) \mathrm{d}t $. – gigo May 07 '21 at 15:52
  • The integral without the $x^2$ term is evaluated analytically in this post. Thus $K(c)=\int dx \ [\operatorname{erf}(ax+b)]^2 e^{-cx^2}$ is known. Even though the result in the linked post has an $\operatorname{erfc}$, it may be cast in the form of your integral using the first result. Your integral is then given by $-\partial_c K(c)|_{c=1}$. The result will likely not be pretty. – Sal May 07 '21 at 17:45
  • An integral similar to that, $ \int_{-\infty}^{\infty} x^2 \Phi (a + b x) \phi(x) \mathrm{d}x$, has been posted. https://math.stackexchange.com/questions/2027470/second-moment-of-the-product-of-normal-cdf-pdf – gigo May 10 '21 at 10:58

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