(Solved): Please give an elaborate correct answer. Problem 2 Consider a random sample of size n=10 from a ...

Please give an elaborate correct answer.

Problem 2 Consider a random sample of size $$n=10$$ from a Gamma distribution with density $$f(x;?,?)=\frac{{?}^{?}}{?(?)}{x}^{??1}{e}^{??x}$$ where $$x>0$$ and $$?,?>0$$, as well. 1. Show that this distribution belongs to the (2-parameter) exponential family of densities. Hint: Let $$?=(?,?{)}^{?}$$. A distribution belongs to the (general) exponential family of densities if $$f(x;?)=\{\begin{array}{ll}\exp \left\{{?}_{i=1}^k{c}_i(?)g_i(\overrightarrow{\mathbf{x}})+d(?)+z(\overrightarrow{\mathbf{x}})\right\}& \overrightarrow{\mathbf{x}}?A\\ 0& \text{ otherwise }\end{array}$$ with $$\overrightarrow{\mathbf{x}}={\left(x_1,\dots ,x_n\right)}^{?}$$ and parameters $$?={\left({?}_1,\dots ,{?}_k\right)}^{?}.c_i(?),d(?)$$ are real-valued functions of $$?$$ not depending on $$\overrightarrow{\mathbf{x}}$$ while $$g_i(\overrightarrow{\mathbf{x}}),z(\overrightarrow{\mathbf{x}})$$ are real-valued functions of $$\overrightarrow{\mathbf{x}}$$ not depending on $$?.A?{\mathbb{R}}^n$$ is a range/support which does not depend on $$?$$.