We make $n$ observations $y_{1},\ldots,y_{n}$ , which, given that values of parameters $\alpha,~\beta$ , are independent observations from a $\mathrm{gamma}(\alpha,\beta)$ distribution. Show that the statistics $T_{1},\ T_{2}$ are sufficient for $\alpha,~\beta$ where  

$$
T_{1}=\sum_{i=1}^{n}y_{i}\qquad\qquad\qquad\mathrm{and}\qquad\qquad\qquad T_{2}=\prod_{i=1}^{n}y_{i}.
$$ 