Timeline for Closed-form solution for an integral involving the p.d.f. and c.d.f. of a $N(0,1)$-distributed random variable

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Feb 4, 2021 at 5:34	comment	added	jvdillon		FYI, the condition in the OwenT derivation is wrong. It should be `-(a/w<0)` and not `-(a<=0)`. The conclusion can be derived from 10,010.3 of Owen's paper and using the property $T(u,v) + T(uv,1/v) = \frac{1}{2} (\Phi(u) + \Phi(uv)) - \Phi(u) \Phi(uv) - \frac{1}{2} [v<0]$.
Oct 22, 2017 at 2:21	comment	added	Student1981		I appreciate your observation, @MattF. I believe I did not choose well my words when I wrote "closed-form solution". I will be more careful next time. However, for my purposes, the approach of Stéphane Laurent's answer is good enough.
Oct 22, 2017 at 1:03	comment	added	user44143		These are efficient reductions of the integral to standard libraries. But I'd say a solution with the bivariate normal CDF or OwenT is not in closed form; I would restrict the term "closed-form" to quantities that can be calculated with at most a single integral of elementary functions.
Oct 21, 2017 at 10:18	comment	added	Stéphane Laurent		@Student1981 I've just added a speed comparison in my answer. `OwenT` is the way to go.
Oct 21, 2017 at 10:17	history	edited	Stéphane Laurent	CC BY-SA 3.0	benchmark
Oct 21, 2017 at 2:06	vote	accept	Student1981
Oct 20, 2017 at 16:58	history	edited	Stéphane Laurent	CC BY-SA 3.0	added 245 characters in body
Oct 20, 2017 at 16:12	history	answered	Stéphane Laurent	CC BY-SA 3.0