The real-complex normal distribution

An expression is derived for the distribution of a mixture of real and complex normal variates. The asymptotic distribution of the resulting real-complex maximum-likelihood estimates is the real-complex normal distribution derived. The covariance matrix of this distribution is particularly important. It is the asymptotic covariance matrix for maximum-likelihood estimates and the Cramer-Rao lower bound on the variance of the real-complex estimates in general. From this covariance matrix, the variance of the reconstructed complex-valued exit wave then follows using the pertinent propagation formulas. The resulting expressions show the dependence of the variance on the free microscope parameters used for experimental design.