Summary: We consider the problem of discriminating between two independent multivariate normal populations,

${N}_{p}(\mu ,{{\Sigma}}_{1})$ and

${N}_{p}(\mu ,{{\Sigma}}_{2})$ having distinct mean vectors

${\mu}_{1}$ and

${\mu}_{2}$ and distinct covariance matrices

${{\Sigma}}_{1}$ and

${{\Sigma}}_{2}$. The parameters

${\mu}_{1}$,

${\mu}_{2}$,

${{\Sigma}}_{1}$,

${{\Sigma}}_{2}$ are unknown and are estimated by means of independent random training samples from each population. We derive a stochastic representation for the exact distribution of the ‘plug-in’ quadratic discriminant function for classifying a new observation between the two populations. The stochastic representation involves only the classical standard normal, chi-square, and

$F$ distributions and is easily implemented for simulation purposes. Using Monte Carlo simulation of the stochastic representation we provide applications to the estimation of misclassification probabilities for the well-known iris data studied by Fisher [Ann. Eugen. 7, 179–188 (1936)]; a data set on corporate financial ratios provided by

*R. A. Johnson* and

*D. W. Wichern* [Applied Multivariate Statistical Analysis, 4th ed., Prentice-Hall, Englewood Cliffs, NJ (1998), see

Zbl 0745.62050]; and a data set analyzed by Reaven and Miller [Diabetologia 16, 17–24 (1979)] in a classification of diabetic status. For part I see J. Multivariate Anal. 77, No. 1, 21–53 (2001;

Zbl 1098.62517).