Consider the problem of discriminating, on the basis of random "training" samples, between two independent multivariate normal populations, Np(Âµ, E1) and Np(Âµ, E2), which have a common mean vector Âµ and distinct covariance matrices E1 and E2. We derive stochastic representations for the exact distributions of the "plug-in" quadratic discriminant functions for classifying a newly obtained observation. For some special values of p, E1 or E2 we obtain explicit formulas and inequalities for the probabilities of misclassification. The stochastic representations for the plug-in discriminant functions involve only chi-squared and F-distributions, hence these representations permit highly efficient simulation of the discriminant functions and estimation of the corresponding probabilities of misclassification. We apply these results to data given by Stocks [Ann. Eugen. 5 (1933):1-55] in a biometric investigation of the physical characteristics of twins; and to data collected by Reaven and Miller [Diabetologia, 16 (1979):17-24] in an investigation of the relationship between various measures of blood chemistry and diabetic status. For each application we estimate the exact probabilities of misclassification, and in the case of Stocks' data we make extensive comparisons with previously published estimates.