“Average-Value Paradox” of the Lognormal Distribution

Abstract

This letter reports seemingly paradoxical behavior of the average of a lognormal distribution. The authors consider a simple stochastic process given by multiplication of uniform random variables. The central limit theorem ensures that a lognormal distribution is asymptotically a good approximation to the actual distribution. Nevertheless, the average of this approximate distribution deviates from the true value. As a solution to this conflict, the authors clarify that the average is generally not preserved under a nonlinear transformation of a random variable, and this causes the disagreement of the lognormal approximation. Instead of the average, the median is proved to be compatible with such a transformation.