Product Moments of Sample Variances and Correlation for Variables with Bivariate Normal Distribution

A general result to obtain the product moments of two sample variances and the sample correlation when the data follow a bivariate normal distribution is derived; the result is expressed in terms of the hypergeometric function. As corollaries, two general equations are stated, one to obtain the moments of the correlation sample and one to obtain the moments of the ratio of two sample variances. To evaluate the product moments in short closed forms, three theorems have been established. The results are used to obtain the expectation and variance for the ratio of two correlated sample variances. Finally, some examples of particular product moments are provided and some validations were carried out.