The uncertainty in the variables and functions in computer simulations can be quantified by probability distributions and the correlations between the variables. We augment the standard computer arithmetic operations and the interval arithmetic approach to include probability distribution variable (PDV) as a basic data type. Probability distribution variable is a random variable that is usually characterized by generalized probabilistic discretization. The correlations or dependencies between PDVs that arise in a computation are automatically calculated and tracked. These correlations are used by the computer arithmetic rules to achieve the convergent approximation of the probability distribution function of a PDV and to guarantee that the derived bounds include the true solution. In many calculations, the calculated uncertainty bounds for PDVs are much tighter than they would have been had the dependencies been ignored. We describe the new PDV arithmetic and verify the effectiveness of the approach to account for the creation and propagation of uncertainties in a computer program due to uncertainties in the initial data.