= Solution
An <exponential dispersion family> has density or mass function, relative to a fixed measure,
$$
f(y;\theta,\phi)=\exp\left\{\frac{y\theta-b(\theta)}{\phi}+c(y,\phi)\right\},\qquad \phi>0.
$$
Here $\theta$ is the <natural parameter>, $b$ is the <cumulant function>, and $\phi$ is the <dispersion parameter>. Assume the <natural parameter> lies in the interior of its domain and derivatives can pass through the normalizing integral. Differentiating normalization once and twice gives
$$
\boxed{\mu=E(Y)=b'(\theta),\qquad
\operatorname{Var}(Y)=\phi b''(\theta)=\phi V(\mu).}
$$
The <variance function> is $V(\mu)=b''((b')^{-1}(\mu))$, where the mean-to-natural-parameter inverse exists. The <dispersion parameter> may be fixed, as in a <Poisson distribution>, rather than estimated.
Back to article page
