1. The normal distribution with fixed variance is NEF-QVF because the variance is constant. The variance can be written , so variance is a degree 0 function of the mean. 2. The Poisson distribution is NEF-QVF because all Poisson distributions have variance equal to the mean , so variance is a linear function of the mean. 3. The Gamma distribution is NEF-QVF because the mean of the Gamma distribution is and the variance of the Gamma distribution is , so the variance is a quadratic function of the mean. 4. The binomial distribution is NEF-QVF because the mean is and the variance is which can be written in terms of the mean as 5. The negative binomial distribution is NEF-QVF because the mean is and the variance is 6. The (not very famous) distribution generated by the generalized hyperbolic secant distribution (NEF-GHS) has and Properties of NEF-QVF The properties of NEF-QVF can simplify calculations that use these distributions. 1. Natural exponential families with quadratic variance functions (NEF-QVF) are closed under convolutions of a linear transformation. That is, a convolution of a linear transformation of an NEF-QVF is also an NEF-QVF, although not necessarily the original one. Given independent identically distributed (iid) with distribution from a NEF-QVF. A convolution of a linear transformation of an NEF-QVF is also an NEF-QVF. Let be the convolution of a linear transformation of X. The mean of Y is . The variance of Y can be written in terms of the variance function of the original NEF-QVF. If the original NEF-QVF had variance function then the new NEF-QVF has variance function where 2. Let and be independent NEF with the same parameter θ and let . Then the conditional distribution of given has quadratic variance in if and only if and are NEF-QVF. Examples of such conditional distributions are the normal, binomial, beta, hypergeometric and geometric distributions, which are not all NEF-QVF. 3. NEF-QVF have conjugate prior distributions on μ in the Pearson system of distributions (also called the Pearson distribution although the Pearson system of distributions is actually a family of distributions rather than a single distribution.) Examples of conjugate prior distributions of NEF-QVF distributions are the normal, gamma, reciprocal gamma, beta, F-, and t- distributions. Again, these conjugate priors are not all NEF-QVF. 4. If has an NEF-QVF distribution and μ has a conjugate prior distribution then the marginal distributions are well-known distributions.