The Variance of the Product ofKRandom Variables

Abstract An exact formula for the variance of the product of K random variables, x 1, x 2, ···, x K , is given as a function of the means and the central product-moments of the xi . The usual approximate variance formula for is compared with the exact formula; e.g., we note, in the case where the x i are mutually independent, that the approximate variance is too small, and that the relative accuracy of the approximation depends on the magnitude of the coefficients of variations of the K random variables. The case where the x i need not be mutually independent is also studied, and variance formulas are presented for two different “product estimators.” (The relative magnitude of the variances of these estimators is shown to be related to the relative accuracy of the approximate variance formula for as compared with the exact variance formula.) When the x i are mutually independent, simple unbiased estimators of these variances are suggested; in the more general case, consistent estimators are presented.