 Mixture allocation indicator

Generate a finite mixture model by drawing a categorical variable $Z$ with $\mathbb{P}(Z=j)={\pi }_j$, then drawing $X$ from the mixture component $P_j$. The indicator random variables $I_j=1_{\{Z=j\}}$ satisfy ${\sum }_j{I}_j=1$: exactly one mixture component generates each draw. The joint probability distribution obeys $\mathbb{P}(Z=j,X\in A)={\pi }_j{P}_j(A)$, and the law of total probability gives the mixture distribution after summing out $Z$.^[ref]

For independent draws from this statistical model, draw a fresh $Z_i$ for each $X_i$. Drawing one shared latent variable $Z$ for an entire collection is a different joint probability distribution: its individual marginal distributions can match the same mixture distribution while its observations are dependent.