Ley y1,…,yn be drawn from a normal distribution with known parameters.
Let x be drawn from another normal distribution with known parameters.
What is the probability that x has rank-k, i.e., it appears in the "k" position of an ordered version of the {y1,…,yn,x} set?
I know that if both distributions have the same parameters, this will give an uniform distribution. If x has a larger mean, it will give an exponential distribution.
However, I am looking for a general formula that, taking into account the Gaussian parameters of both distributions, is able to obtain the above probability for generic normal distributions.