Probability that values from different normal distributions will be in a given order?

#1
I am looking at several normal distributions that describe the same metric for different populations (independent). I want to find the probability that they are in a certain order.

For example, I have distributions A, B, C, and D. I want to know the probability that they are ranked from highest to lowest B,C,A,D (or any other permutation).

I have been trying to research it myself; however, I am not getting to a good answer. I think that this page would have something to do with the solution, but I don't know how to extend it to multiple distributions in a certain order: Probability of a point taken from a certain normal distribution will be greater than a point taken from another?

I have been attempting this for a while and keep getting incorrect results. As a sanity check I would find the probability of all permutations and sum them up. I was looking for them to equal 100%, but the closest I ever got was ~76%.

Any help would be greatly appreciated.
 

BGM

TS Contributor
#2
Re: Probability that values from different normal distributions will be in a given or

Essentially you will need a numerical tool to evaluate the multivariate normal CDF.

Note that \( \Pr\{D < A < C < B\} \)

\( = \Pr\{D < A, A < C, C < B\} \)

\( = \Pr\{D - A < 0, A - C < 0, C - B < 0\} \)

and

\( \begin{bmatrix} D - A \\ A - C \\ C - B \end{bmatrix} \sim \mathcal{N}
\left(\begin{bmatrix} \mu_D - \mu_A \\ \mu_A - \mu_C \\ \mu_C - \mu_B
\end{bmatrix},
\begin{bmatrix} \sigma^2_D + \sigma^2_A & -\sigma^2_A & 0 \\
-\sigma^2_A & \sigma^2_A + \sigma^2_C & -\sigma^2_C \\
0 & -\sigma^2_C & \sigma^2_C + \sigma^2_B \end{bmatrix}\right) \)