Consider a probability measure μ on [0, 1]d, and independent random variables X⃗1, …, X⃗n distributed according to μ. Let Vn = Vn(X⃗1, …, X⃗n) (resp. Rn = Rn(X⃗1, …, X⃗n)) be the minimal number of unit size d-dimensional bins needed to pack items of size X⃗1, …, X⃗n in the vector packing (resp. rectangular packing) problem. For a large class of distributions μ, we characterize v(μ) = limn→∞ Vn/n and r(μ) = limn→∞ Rn/n.