The Six NEF-QVFs
The six NEF-QVF are written here in increasing complexity of the relationship between variance and mean.
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
Read more about this topic: Natural Exponential Family, Natural Exponential Families With Quadratic Variance Functions (NEF-QVF)