Prediction interval is a widely used tool in industrial applications to predict the distribution of future observations. The exact minimum coverage probability and the average coverage probability of the conventional prediction interval for a discrete random variable have not been accurately derived in the literature. In this paper, procedures to compute the exact minimum confidence levels and the average confidence levels of the prediction intervals for a discrete random variable are proposed. These procedures are illustrated with examples and real data applications. Based on these procedures, modified prediction intervals with the minimum coverage probability or the average coverage probability close to the nominal level can be constructed.