Let $$\mathcal {C}=\{C_1,\ldots ,C_n\}$$ C = { C 1 , … , C n } be a set of $$n$$ n pairwise-disjoint convex sets of constant description complexity, and let $$\pi $$ π be a probability density function (density for short) over the non-negative reals. For each $$i$$ i , let $$K_i$$ K i be the Minkowski sum of $$C_i$$ C i with a disk of radius $$r_i$$ r i , where each $$r_i$$ r i is a random non-negative number drawn independently from the distribution determined by $$\pi $$ π . We show that the expected complexity of the union of $$K_1, \ldots , K_n$$ K 1 , … , K n is $$O(n^{1+{\varepsilon }})$$ O ( n 1 + ε ) for any $${\varepsilon }> 0$$ ε > 0 ; here the constant of proportionality depends on $${\varepsilon }$$ ε and the description complexity of the sets in $$\mathcal {C}$$ C , but not on $$\pi $$ π . If each $$C_i$$ C i is a convex polygon with at most $$s$$ s vertices, then we show that the expected complexity of the union is $$O(s^2n\log n)$$ O ( s 2 n log n ) . Our bounds hold in a more general model in which we are given an arbitrary multi-set $$\varTheta =\{\theta _1,\ldots ,\theta _n\}$$ Θ = { θ 1 , … , θ n } of expansion radii, each a non-negative real number. We assign them to the members of $$\mathcal {C}$$ C by a random permutation, where all permutations are equally likely to be chosen; the expectations are now with respect to these permutations. We also present an application of our results to a problem that arises in analyzing the vulnerability of a network to a physical attack.