On the Distribution of a Positive Random Variable Having a Discrete Probability Mass at the Origin*

In a number of situations we are faced with the problem of determining efficient estimates of the mean and variance of a distribution specified by (i) a non-zero probability that the variable assumes a zero value, together with (ii) a conditional distribution for the positive values of the variable. This estimation problem is analyzed and its implications for the Pearson type III, exponential, lognormal and Poisson series conditional distributions are investigated. Two simple examples are given.