the parameterization in the linked article in the Poisson case. The two parameterizations are related by , where λ is the mean parameter, and so that the density may be written as for , so This alternative parameterization can greatly simplify calculations in mathematical statistics. For example, in Bayesian inference, a posterior probability distribution is calculated as the product of two distributions. Normally this calculation requires writing out the probability distribution functions (PDF) and integrating; with the above parameterization, however, that calculation can be avoided. Instead, relationships between distributions can be abstracted due to the properties of the NEF described below. An example of the multivariate case is the multinomial distribution with known number of trials. Properties The properties of the natural exponential family can be used to simplify calculations involving these distributions. Univariate case 1. The cumulants of an NEF can be calculated as derivatives of the NEF's cumulant generating function. The nth cumulant is the nth derivative of the cumulant generating function with respect to t evaluated at t = 0. The cumulant generating function is The first cumulant is The mean is the first moment and always equal to the first cumulant, so The variance is always the second cumulant, and it is always related to the first and second moments by so that Likewise, the nth cumulant is 2. Natural exponential families (NEF) are closed under convolution. Given independent identically distributed (iid) with distribution from an NEF, then is an NEF, although not necessarily the original NEF. This follows from the properties of the cumulant generating function. 3. The variance function for random variables with an NEF distribution can be written in terms of the mean. 4. The first two moments of a NEF distribution uniquely specify the distribution within that family of distributions. Multivariate case In the multivariate case, the mean vector and covariance matrix are where is the gradient and is the Hessian matrix. Natural exponential families with quadratic variance functions (NEF-QVF) A special case of the natural exponential families are those with quadratic variance functions. Six NEFs have quadratic variance functions (QVF) in which the variance of the distribution can be written as a quadratic function of the mean. These are called NEF-QVF. The properties of these distributions were first described by Carl Morris. The six NEF-QVFs The six NEF-QVF are written here in increasing complexity of the relationship between variance and mean.