Discussion
Suppose that the random vector \((X_{1}, X_{2}, X_{3})'\) has a multinomial distribution with parameters \(n\), \(p_{1}\), \(p_{2}\) and \(p_{3}\). Then, which of the following statements is FALSE ?
Answer: A
The false statement is that \(X_{1}\) and \(X_{2}\) are independent binomial variables.
Each margin of a multinomial is binomial, so \(X_{1} \sim B(n, p_{1})\) and \(X_{2} \sim B(n, p_{2})\), but they are not independent: they compete for the same \(n\) trials, and \(X_{1} + X_{2} \le n\). Their covariance is \(-np_{1}p_{2} \neq 0\).
The other statements are true. \(\text{Cov}(X_{i}, X_{j}) = -np_{i}p_{j}\) for \(i \neq j\), so \(\text{Cov}(X_{1}, X_{3}) = -np_{1}p_{3}\). The margin \(X_{3}\) is \(B(n, p_{3})\), with variance \(np_{3}(1 - p_{3})\). And merging categories 1 and 2 gives a binomial with success probability \(p_{1} + p_{2} = 1 - p_{3}\), so \(X_{1} + X_{2} \sim B(n, 1 - p_{3})\).
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