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I am trying to solve the integral of a gaussian cumulative distribution function and a gaussian probability function. On this site I have seen solutions of similar, less general integrals (e.g. Integral of the product of Normal density and cdf) . Does a closed form solution for this specific case exist? $$\int_{s_2=-\infty }^{s_2=y} \phi(s;\mu;\sigma)\times \Phi(r; cs_2;\tau) \text{d}{s} $$ where Φ is the cdf of a gaussian distribution, and ϕ its density.

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  • $\begingroup$ That is too many subscripts for a question here. If you can write this as $$\int \phi(s;\mu,\sigma)\Phi(r;cs,\tau)ds$$ please edit the question like that! $\endgroup$
    – user44143
    Commented Jan 13, 2022 at 15:13
  • $\begingroup$ Do you have a response to the answer on this page? $\endgroup$
    – Iosif Pinelis
    Commented Jan 18, 2022 at 0:50

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Mathematica cannot take such integrals, even for zero-mean normal distributions. So, it is highly unlikely that they can expressed in closed form.

Here is the image of the corresponding Mathematica notebook:

enter image description here

An exceptional trivial case is when the two normal distributions are the same:

enter image description here

That is, here we get $F(y)^2/2=\Phi(y)^2/2$, by the substitution $t=\Phi(x)$.

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