Sheldon Ross 10: Exercise 3.35

Question: Consider n multinomial trials, where each trial independently results in outcome i with probability psuch that Σiki=1 pi = 1. With Xi equal to the number of trials that result in the outcome i, find E[ X1 | X2 > 0].


Analytical Solution

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

E[ X1 ] = E[E[ X1 | X2 ]]

E[X1] = E[X1 | X2 = 0] P{X2 = 0} + E[X1 | X2 > 0] P{X2 > 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.

np1 = E[X1 | X2 = 0] P{X2 = 0} + E[X1 | X2 > 0] P{X2 > 0}

np1 = E[X1](1-p2) (1-p2)n + E[X1 | X2 > 0] (1-(1-p2)n)

np1 = np1(1-p2) (1-p2)n + E[X1 | X2 > 0] (1-(1-p2)n)

E[X1 | X2 > 0] = np(1 – (1-p2)n-1)(1-(1-p2)n)


End of the post 🙂