Bounded bivariate distributions with nearly normal marginals

A number of examples of nonnormal bivariate distributions whose marginal distributions are normal have been described in the literature. Such examples can be extremely useful in classroom discussions of univariate and bivariate normality. The present note introduces a new, practically motivated example involving maximum and minimum daily temperatures, in which the marginal distributions appear to be normally distributed, whereas the bivariate distribution is not. Two differences exist compared to previous examples: (1) there is a practical motivation; (2) despite the apparent normality of the marginal distributions, it can be shown that such normality cannot really be valid, except in an unrealistic special case. The example illustrates a number of additional points that would be of value in a classroom discussion of univariate and bivariate distributions.