Multinomial aggregation ($M_j={\sum }_{i\in G_j}{N}_i,q_j={\sum }_{i\in G_j}{p}_i$)

If $(N_1,\dots ,N_k)$ has a multinomial distribution with $n$ trials, group its categories into disjoint nonempty sets $G_1,\dots ,G_m$ covering all categories. The grouped counts in the displayed equation have the multinomial distributionEach original trial belongs to exactly one group, its group probability is the sum of its category probabilities, and grouping preserves independence between trials. This gives the rule without summing the original probability mass function over every possible internal count pattern.

For the four-trial example with probabilities $(1/2,3/10,1/5)$, merge the last two categories. Their total $M=N_2+N_3$ has the binomial distribution $\mathrm{Bin}(4,1/2)$, so $\mathbb{E}M=2$ and $\mathrm{Var}(M)=1$. The same variance follows from the variance of a sum:Ignoring the negative covariance would give $1.48$, although the two counts share a fixed total. Multinomial aggregation lets the resolution of category labels change while retaining a consistent model for the counts.