We investigate Mahonian and Eulerian probability distributions given by inversions and descents in general finite Coxeter groups. We provide uniform formulas for the means and variances in terms of Coxeter group data in both cases. We also provide uniform formulas for the double-Eulerian probability distribution of the sum of descents and inverse descents. We finally establish necessary and sufficient conditions for general sequences of Coxeter groups of increasing rank under which Mahonian and Eulerian probability distributions satisfy central and local limit theorems.