Instead of my general formulation below, I will give an example:
Inside a 2x3x4 inch match box is a female insect F and a male insect M. F is at a corner of the matchbox and M is at a random point. The matchbox has random, stationary "stinky pockets" floating around occupying 1/3 of the total match box. The male is 4 times more likely to be floating around in a "stinky pocket" than in the remaining room of the matchbox, and we don't know how the stinky pockets are assigned in the space within the box.
Now consider a variation of where there are no stinky pockets in the match box (M is equally likely to be anywhere).
Are the variances of the rectilinear distance that F has to cover to reach M the same in the "stinky patches" and in the normal situation?
My intuition tells me yes, because the probability of finding M at each point remains the same.
Consider any probability distribution f(x). The support/domain-assignment remains the same, but the y-values/probabilities (weight) are permuted in a random fashion. We do not know the new permutation. Is the variance the same as in the original distribution?
In other words, if we consider all possible permutations (give each one equal weight), on average is the variance the same as in the original distribution?