Let $$X = (X_{1},..., X_{n})$$ consist of iid random variables s.t. $$X_{i}\sim Bin({m,\theta }),$$ for known$$m \in \mathbb{N},$$ and unknown$$ \theta \in (0,1)$$
Describe statistical model including $$ S, f_{\theta},\Theta $$
So far I have:
$$S = [0,1,...,m]^{n}$$ $$\Theta=(0,1)$$ $$f_{\theta_{i}}= \begin{pmatrix} m \\x_{i} \end{pmatrix} \theta^{x_{i}}(1-\theta)^{m-x_{i}}$$ $$f_{\theta}=\prod_{i=1}^{n}\begin{pmatrix} m \\x_{i} \end{pmatrix} \theta^{x_{i}}(1-\theta)^{m-x_{i}}$$
not sure how to simplify last expression
So far so good. Firstly we can use that $\prod\limits_{i=1}^{n} f(i)\cdot g(i)=\prod\limits_{i=1}^{n} f(i)\cdot \prod\limits_{i=1}^{n} g(i)$, where $f(i), g(i)$ are arbitrary terms which depends on index $i$.
$$f_{\theta}=\prod_{i=1}^{n}\begin{pmatrix} m \\x_{i} \end{pmatrix} \theta^{x_{i}}(1-\theta)^{m-x_{i}}$$
$$=\prod_{i=1}^{n}\begin{pmatrix} m \\x_{i} \end{pmatrix} \cdot \prod_{i=1}^{n} \theta^{x_{i}}\cdot \prod_{i=1}^{n}(1-\theta)^{m-x_{i}}$$
Then $\prod_{i=1}^{n} \theta^{x_{i}}=\theta^{x_{1}}\cdot \theta^{x_{2}}\cdot \ldots \cdot \theta^{x_{n}}=\theta^{\, \sum\limits_{i=1}^n x_i} $
$$=\prod_{i=1}^{n}\begin{pmatrix} m \\x_{i} \end{pmatrix} \cdot \theta^{\, \sum\limits_{i=1}^n x_i}\cdot (1-\theta)^{\, \sum\limits_{i=1}^n (m-x_i)}$$
$$=\theta^{\, \sum\limits_{i=1}^n x_i}\cdot (1-\theta)^{\, \sum\limits_{i=1}^n (m-x_i)}\cdot \prod_{i=1}^{n}\begin{pmatrix} m \\x_{i} \end{pmatrix} $$