THE ANALYSIS OF THE VARIANCE AND COVARIANCE OF PRODUCTS

General expressions for the covariance of a product of n random variables with a product of m random variables, and for the expectation of a product of random variables, in terms of the means and central product-moments of the constituent variables, are derived. A general method for the partition of the variance and covariance of products of random variables into terms due to variation in single variables, variation in and covariation between pairs of variables, triples, etc. is proposed. This would be of value in the interpretation of many statistical analyses involving products or sums of products of random variables. Three examples are given in which the results are used to help identify the relative importance of, and the effects of relationships between, the factors in lifetime reproductive success and in the magnitude of fluctuations of population density, and in microvillus growth in animal physiology.