**Question: **Consider n multinomial trials, where each trial independently results in outcome i with probability p_{i }such that Σ_{i}k_{i}=1 p_{i} = 1. With X_{i} equal to the number of trials that result in the outcome i, find E[ X_{1} | X_{2} > 0].

**Analytical Solution**

Since an expectation is being asked for here, we can expand the E[X] to its conditional form.

E[ X_{1} ] = E[E[ X_{1} | X_{2} ]]

E[X_{1}] = E[X_{1} | X_{2} = 0] P{X_{2} = 0} + E[X_{1} | X_{2} > 0] P{X_{2} > 0}

Even though we know that the system is multinomial one, the conditionality causes reduction of the space and we end up with a binomial like situation. All the subsequent quantities are based off of this space reduction.

np_{1} = E[X_{1} | X_{2} = 0] P{X_{2} = 0} + E[X_{1} | X_{2} > 0] P{X_{2} > 0}

np_{1} = ^{E[X1]}⁄_{(1-p2)} (1-p_{2})^{n} + E[X_{1} | X_{2} > 0] (1-(1-p_{2})^{n})

np_{1} = ^{np1}⁄_{(1-p2)} (1-p_{2})^{n} + E[X_{1} | X_{2} > 0] (1-(1-p_{2})^{n})

E[X_{1} | X_{2} > 0] = ^{np1 (1 – (1-p2)n-1)}⁄_{(1-(1-p2)n)}

End of the post 🙂