Probability of Ranking Suppose that I have 3 normal distributions (I wish to extend this to K), such that $p_k\sim\mathcal{N}(\mu_k,\sigma_k^2)$. Assume that $(\mu_k,\sigma_k^2)$ are known. How would you go about calculating the probability that:
$p(p_1<p_2<p_3)$? 
I understand that when k=2, we come down to the case of the elo model, where we have to use the CDF of a normal distribution (of the for $\Phi(\frac{\mu_1-\mu_2}{\sqrt{2}\sigma})$. I'm not quite sure how this pans out for more than two players.
Side question, if there is no closed form solution, is anyone aware of approximations to this (eg. Copulas maybe?).
 A: In the case $K=3$ this probability can be reduced to a one-dimensional integral, which can maybe be solved analytically, but at least numerically. For larger $K$ this is similar to integrals occurring with multinomial probit models, in particular you could look at the references given in the help for mnp in R package MNP. But for $K=3$:
$$\DeclareMathOperator{\P}{\mathbb{P}}
\P(p_1 < p_2 < p_3)=\int_{-\infty}^\infty \P(p_1 < p_2 <p_3 \mid p_2=u)\phi(\frac{u-\mu_2}{\sigma_2})\frac1{\sigma_2}\; du$$ and by independence we can simplify these (details not given) to 
$$
\int_{-\infty}^\infty \Phi(\frac{u-\mu_1}{\sigma_1})(1-\Phi(\frac{u-\mu_3}{\sigma_3}))\phi(\frac{u-\mu_2}{\sigma_2})\frac1{\sigma_2}\; du
$$ Here $\phi, \Phi$ are standard normal density and cdf respectively. 
A numerical example with R: 
s1 <- s3 <- 1.;  mu1 <- mu2 <- 10; mu3 <- 9; s2 <- 2
integrand <- function(u) pnorm((u-mu1)/s1)*pnorm((u-mu3)/s3, lower.tail=FALSE)*dnorm((u-mu2)/s2)/s2

integrate(integrand, lower=-Inf, upper=Inf)
0.03572245 with absolute error < 4.3e-05

