integral involving the normal CDF Does anyone know if I have a chance to get a closed form for the following integral (and if yes, what would be the trick)?
$$\int \left\{1-\Phi(x)\right\}^k \Phi(x)^{n-k} \varphi\left(\frac{x-\mu}{\sigma}\right) dx,$$
where $\Phi$ and $\varphi$ are the normal cumulative distribution and density functions, $k \leq n$ are integers, and $\mu$ and $\sigma >0$ are real numbers.
I had a look at this page https://en.wikipedia.org/wiki/List_of_integrals_of_Gaussian_functions,
but I'm not sure it contains what I need.
 A: One better reference for Gaussian integrals is "A table of normal integrals" by D. B. Owen, in Commun. Statist. Simulat. Computa. B9(4), 389-419, 1980.  That reference includes a few very special cases of your integral, but nothing close to its generality. One very special case which yields simply is when $\mu=0, \sigma=1$ when the substitution $u=\Phi(x)$ reduces the integral to
$$
   \int_0^1 u^{n-k} (1-u)^k \; du = \frac{k(n-k)B(k,n-k)}{n(1+n)}
$$
where $B$ is the beta function.  That could at least serve as a check on numerical solutions. 
A: Let $X_1, X_2, \ldots, X_n, Y$ be independent normal random variables with $X_i \sim N(0,1)$ for $i = 1,2,\ldots, n$, and $Y \sim N(\mu, \sigma^2)$. Then, consider the probability that $k$ specified $X_i$ are larger than $Y$ and the remaining $n-k$ of the $X_i$ are smaller than $Y$. As a specific example, suppose that we wish to find the probability of the event 
$$\mathcal A =\big(X_1 > Y, X_2 > Y, \ldots, X_k > Y,  X_{k+1} < Y, X_{k+2} < Y, \ldots, X_ n < Y\big).$$  Following the argument used in this answer of mine, we have that
the conditional probability of $\mathcal A$ given that $Y = x$ is
\begin{align}P(\mathcal A \mid Y = y) &= P(X_1>x, \ldots, X_k > x, X_{k+1} < y, \ldots, X_n < y \mid Y =x)\\&= P(X_1>x, \ldots, X_k > x, X_{k+1} < y, \ldots, X_n < y)\\
&= \prod_{i=1}^k P(X_i > y)\prod_{i=k+1}^n P(X_i < y)\\
&=[1-\Phi(x)]^{k}[\Phi(x)]^{n-k}
\end{align}
and so
$$P(\mathcal A) = \int_{-\infty}^\infty
[1-\Phi(x)]^{k}[\Phi(x)]^{n-k} \phi\left(\frac{x-\mu}{\sigma}\right)\,\mathrm dx.$$
There is no closed-form expression for this probability and there are no simple approximations that can be used except in special cases such as $k=0$ (cf. @whuber's comment).
