Variants 1 and 2 are not actually standard exponential families at all. Rather they are "curved exponential families", i.e. there are formula_79 independent parameters embedded in a formula_80-dimensional parameter space. Many of the standard results for exponential families do not apply to curved exponential families. An example is the log-partition function formula_89, which has the value of 0 in the curved cases. In standard exponential families, the derivatives of this function correspond to the moments (more technically, the cumulants) of the sufficient statistics, e.g. the mean and variance. However, a value of 0 suggests that the mean and variance of all the sufficient statistics are uniformly 0, whereas in fact the mean of the formula_90th sufficient statistic should be formula_91. (This does emerge correctly when using the form of formula_89 shown in variant 3.)