Considering the characteristics of the bivariate normal distribution, in which uncorrelation of two random variables is equivalent to their independence, it is interesting to verify this problem in other distributions. In other words, whether the multivariate normal distribution is the only distribution in which uncorrelation is equivalent to independence. In this paper, we answer to this question and establish generalized Farlie-Gumbel-Morgenstern (FGM) family is another family of distributions under which uncorrelation is equivalent to independence.
Barmalzan, G. and Vali, F.
Farlie-Gumbel-Morgenstern Family: Equivalence of Uncorrelation and Independence,
Applications and Applied Mathematics: An International Journal (AAM), Vol. 12,
2, Article 8.
Available at: https://digitalcommons.pvamu.edu/aam/vol12/iss2/8