Let $(X_1,X_2,...,X_n)$ be a random sample from a distribution with discrete probability $f_{\theta,j}$, where $\theta \in (0,1)$, $j=1,2$, $f_{\theta,1}$ is the Poisson distribution with mean $\theta$, and $f_{\theta,2}$ is the binomial distribution with size $1$ and probability $\theta$. Find a two-dimensional minimal sufficient statistic for $(\theta,j)$.
Approach:
The joint distribution of $X$ is given as $$f_{\theta}(x) = \left[ \frac{e^{- \theta} \theta ^{\sum X_i}}{\prod_{i=1}^{n} X_i !} I_{\{j=1\}}. \theta^{\sum X_i}(1-\theta)^{n-\sum X_i} I_{\{j=2\}} \right]$$ which can be re-arranged as: $$\left[ \frac{e^{-\theta}}{\prod_{i=1}^{n} x_i !} I_{\{j=1\}} (1-\theta)^n I_{\{j=2\}} \right]. \exp\{ \sum X_i \log \theta. I_{\{j=1\}} + \sum X_i \log \left( \frac{\theta}{1-\theta} \right) I_{\{j=2\}} $$.
Seems like $\sum X_i$ is minimal sufficient but turns out it is not sufficient for $(\theta,j)$. Intuitively, we should have a two-dimensional statistic to minimal sufficient but I don't have a clue about how to find it.