The generalized Kneser-Poulsen conjecture predicts that the volume of the union/intersection of some balls in the Euclidean, spherical or hyperbolic space does not decrease/increase when the balls are rearranged so that the distances between the centers increase. First we overview different weighted versions of this conjecture then turn our attention to inequalities on the expectation of the volume of the union/intersection of balls with fixed centers and random radii.