They are only 6 exponential family distributions that admit variance as a quadratic function (QVF=quadratic variance function) of the parameter. For the multivariate case, it is a bit more complex but well defined and studied:
The $2d+4$ simple quadratic natural exponential families on $R^d$
They are only 6 exponential family distributions that admit variance as a quadratic function (QVF=quadratic variance function) of the parameter. For the multivariate case, it is a bit more complex but well defined and studied:
The $2d+4$ simple quadratic natural exponential families on $R^d$
Frank.