# Expected ranked position of a set of normal-distributed random variables

• Mar 27th 2011, 09:12 AM
csiko
Expected ranked position of a set of normal-distributed random variables
Given several normal-distributed random variables X1 .. Xn with means m1,m2,m3 .. mn and variance s1,s2,s3 .. sn respectively, how can the ranking (when sorted from highest to lowest) of a particular sample can be predicted in general?

For example, if X1 ~ N(1,1) , X2 ~ N(2,1) and X3 ~ N(1,2), what will be the average "ranked position" (after sorting) of the X1 sample, the X2 sample and the X3 sample?

I tried to research the problem, but all I came up was a solution for the case with two variables.
Thanks in advance for help and hints!
• Mar 27th 2011, 04:21 PM
theodds
This probably isn't the most efficient way of doing this, but I guess you can get the probability of each of the orderings and work with that (i.e. \$\displaystyle P(X_1 \le X_2 \le X_3)\$ and so forth, which I assume you know how to get). Probably that would get out of control for large n though. At the very least you can streamline those sort of calculations - you would be working with a bunch of multivariate normals - but then you would have to do \$\displaystyle n!\$ of these.

I don't see any reason to think this is an easy problem in practice if you really want to get exact answers. You could Monte Carlo this if it is a practical problem.

At least this takes care of the example you gave though. You only need to calculate 6 probabilities to know the distributions of the rankings.