Let $f_{0}$ be a probability density function, with cumulant generating function $K$. Define what it means for a random variable $Y$ to have a model function of exponential dispersion family form, generated by $f_{0}$. Compute the cumulant generating function $K_{Y}$ of $Y$ and deduce expressions for the mean and variance of $Y$ that depend only on first and second derivatives of $K$.