Another important property of the multinomial distribution is that its conditional distributions are also multinomial. For example,
Conditional distributions
If \((X_1, X_2,\dots, X_g)\) have a \(\MultinomDistn(n, \pi_1, \dots, \pi_g)\) distribution, the conditional distribution of \((X_2, \dots, X_g)\), given that \(X_1 = x_1\) is \(\MultinomDistn(n-x_1, \pi_2^*, \dots, \pi_g^*)\) where
\[ \pi_i^* \;\;=\;\; \frac{\pi_i}{1 -\pi_1} \](Proved in full version)
Question
Consider an opinion poll in which the three possible responses have probabilities
P(Agree) = 0.3,
P(Neutral) = 0.4
P(Disagree) = 0.3
If a random sample of \(n\) = 4 people is asked and the numbers agreeing, neutral and disagreeing are \(X_A\), \(X_N\) and \(X_D\), then
\[ (X_A, X_N, X_D) \;\sim\; \MultinomDistn(n=4, 0.3, 0.4, 0.3) \]If it is known that one person in the sample Agrees, i.e. \(X_A = 1\), what is the distribution of the number who are neutral?
(Solved in full version)