Let $X_1,X_2,\ldots,X_n$ be i.i.d random variables with pmf $f_{\theta,j}(\cdot)$ where $\theta \in (0,1)$ and $j=1,2$.
$f_{\theta,j}:$ pmf of Poisson $(\theta)$ when $j=1$ and $f_{\theta,j}:$ pmf of Geometric $(\theta)$ when $j=2$. Find a sufficient statistic for $(\theta,j)$. Is it minimal sufficient?
I am really confused how to write the joint pmf for $f_{\theta,j}$ .
You want to write function which is equal to $f_1=f_{Poiss(\theta)}(x_1,\ldots,x_n)$ for $j=1$ and to $f_2=f_{Geom(\theta)}(x_1,\ldots,x_n)$ for $j=2$. Say, you can use indicators: $$ f=f_1\mathbf 1_{\{j=1\}}+f_2\mathbf 1_{\{j=2\}}. $$ And since you need to find sufficient statistic, this form in not suitable. It is better for Neyman-Fisher factorization theorem that joint density is a product. So, you can rewrite the equality above as $$ f=f_1^{\mathbf 1_{\{j=1\}}}\times f_2^{\mathbf 1_{\{j=2\}}} = f_1^{2-j}\times f_2^{j-1}. $$ Check that for $=1,2$ $$ \mathbf 1_{\{j=1\}} = 2-j, \quad \mathbf 1_{\{j=2\}} = j-1 $$ and find joint pmf in the form $$f_{\theta,j}(x_1,\ldots,x_n)=\left(f_{Poiss(\theta)}(x_1,\ldots,x_n)\right)^{2-j}\times \left(f_{Geom(\theta)}(x_1,\ldots,x_n)\right)^{j-1}$$