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

Let $\phi(\cdot)$ and $\Phi(\cdot)$ be the probability and cumulative density functions, respectively, of a random variable with distribution $\text{N}(0,\,1)$. I was wondering if you could help me to compute
$$\int_{-\infty}^{w}\phi(x)\cdot\Phi(a+b\cdot x)\,\text{d}x\mbox{,}$$ where $(a,\,b,\,w)\in\mathbb{R}^3$ and $b\neq 0$, please. Thanks a lot for your help.

• I find this one and delete my answer, when i read comments, I cannot help laughing, math.stackexchange.com/questions/381308/… The idea is to express $e^{−x^2/2}$as Maclaurin series, I think – Oct 20, 2017 at 12:30
• Hi, @DeepNorth. I am looking for a closed-form expression for the integral in the question because integrals decrease the efficiency of an algorithm I am programming in R-software. After examining the answer provided in the link, I do not know whether it will contribute to improve the efficiency of my algorithm. I will have a try. Thanks. Oct 20, 2017 at 12:40
• An appropriately chosen series may converge extremely rapidly. Alternatively, consider a saddlepoint approximation: this kind of integral is particularly amenable to that approach.
– whuber
Oct 20, 2017 at 13:25
• Thanks for the suggestion, @whuber. I will spend some time learning about saddlepoint approximations before approximating the integral in the question. Oct 20, 2017 at 13:37
• Hi, nice people. This question was answered here. Oct 22, 2017 at 7:33