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.

  • $\begingroup$ 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 – $\endgroup$
    – Deep North
    Commented Oct 20, 2017 at 12:30
  • $\begingroup$ 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. $\endgroup$ Commented Oct 20, 2017 at 12:40
  • $\begingroup$ An appropriately chosen series may converge extremely rapidly. Alternatively, consider a saddlepoint approximation: this kind of integral is particularly amenable to that approach. $\endgroup$
    – whuber
    Commented Oct 20, 2017 at 13:25
  • $\begingroup$ Thanks for the suggestion, @whuber. I will spend some time learning about saddlepoint approximations before approximating the integral in the question. $\endgroup$ Commented Oct 20, 2017 at 13:37
  • $\begingroup$ Hi, nice people. This question was answered here. $\endgroup$ Commented Oct 22, 2017 at 7:33