Define an exponential dispersion family. Prove that the range of the natural parameter, $\Theta$, is an open interval. Derive the mean and variance as a function of the log normalizing constant.

[Hint: Use the convexity of $e^{x}$, i.e. $e^{p x+(1-p) y} \leqslant p e^{x}+(1-p) e^{y}$ for all $\left.p \in[0,1] .\right]$