The Gamma distribution with shape parameter $\alpha>0$ and scale parameter $\lambda>0$ has probability density function

$$f(y ; \alpha, \lambda)=\frac{\lambda^{\alpha}}{\Gamma(\alpha)} y^{\alpha-1} e^{-\lambda y} \quad \text { for } y>0$$

Give the definition of an exponential dispersion family and show that the set of Gamma distributions forms one such family. Find the cumulant generating function and derive the mean and variance of the Gamma distribution as a function of $\alpha$ and $\lambda$.