THE CONVOLUTION OF THE NORMAL AND LOGNORMAL DISTRIBUTIONS

Let Z be a three-parameter lognormal variate, U a normal with zero mean and define X = Z + U. The marginal distribution of X is then the convolution of the lognormal with the normal - a distribution we will abbreviate to LNN. Expressions for the density and distribution function of the LNN are given, and its properties sketched. Maximum likelihood and method of moments estimators of the parameters of the LNN are given. Calibration requires the conditional distribution of X given Z. This is derived, along with the calibration curve E(X\Z) and its standard error function. An application to X-ray fluorescence counting of in situ gold grades is discussed.