I have several *positive* random variables $x_i,\ i=1,...,N$ taken from different unknown distributions (these distributions can be closely approximated by log-normal if needed). I can sample these variables as many times as needed; all variables are sampled simultaneously.

From each sample my software selects the minimal variable $x_j$ and outputs its index $j$.

The problem is: all values of output index $1 \leq j \leq N$ should have equal probabalities. So, random variables should be somehow adjusted to be "equal" for the minumum operation.

I decided to multiply each random variable by some constant $k_i$. First idea was that $k_i=1/E[x_i]$, so that all adjusted variables will have equal expected values. The output indices indeed became much more uniform, but I pretty sure that my solution is wrong.

The software should also dynamically adapt to gradually changing distributions of random variables (noise shape of sensors depend on temperature).

Do you have any advice?