# The probability that several normally distributed random variables have a particular order

Let's say we have n random, independent variables $X_1,\ldots,X_n$ with normal distributions. Is there a reasonable way how to compute the probability that they have a particular order, for example

$$P(X_1\leq X2 ~\land~ X2\leq X3 ~\land~ \ldots ~\land~ X_{n-1}\leq X_n)$$

?

The canonical way would be to compute the integral

$$\int_{-\infty}^{\infty} \int_{x_1}^{\infty} \ldots\int_{x_{n-1}}^{\infty} f_1(x_1) f_2(x_2) \ldots f_n(x_n) \mathrm{d}x_n \ldots \mathrm{d}x_2\mathrm{d}x_1$$

but this seems not to be that easy, and I hope there is a better way, perhaps taking advantage of the fact that they are normal distributions.

To be more specific, we have several sampled data sets that are expected to have normal distribution. Currently we estimate the parameters of their normal distribution and then approximate the probability using the Monte Carlo method.

• Are there any relationships among the parameters of those distributions, such as having a common mean or common variance? – whuber Jul 28 '15 at 18:57
• @whuber No, we don't expect any such relationships - both the means and variances are different in general. – Petr Pudlák Jul 28 '15 at 18:59
• Then you're out of luck and you'll need to compute that integral numerically. For a glimpse at what can happen, see the simulation I reported at stats.stackexchange.com/a/61463 (in a scatterplot matrix). Although those were not Normal variates, similar things can happen with Normal distributions, too. – whuber Jul 28 '15 at 19:03