Multinomial distribution

A random vector $⃗ 𝐗 : 𝜉 \to ℝ 𝑘$ has a $𝑘$ -dimensional multinomial distribution iff it is the sum of $𝑛$ independently distributed categorically distributed variables #m/def/prob

𝑌 𝑖 \sim C a t 𝑘 (⃗ 𝐩) 𝑋 = 𝑛 \sum 𝑖 = 1 𝑌 𝑖 𝑋 \sim M u l t i 𝑘 (𝑛, ⃗ 𝐩)

ℙ (⃗ 𝐗 = ⃗ 𝐱) = 𝑛! 𝑘 \prod 𝑖 = 1 𝑝 𝑥 𝑖 𝑖 𝑥 𝑖!

and $𝑋 𝑖 \sim B i n (𝑛, 𝑝 𝑖)$ . Hence this generalizes the binomial distribution.

Properties

Multinomial lumping $(𝑋 1 + 𝑋 2, 𝑋 3, \dots, 𝑋 𝑘) \sim M u l t 𝑘 - 1 (𝑛, (𝑝 1 + 𝑝 2, 𝑝 3, \dots, 𝑝 𝑘))$
Multinomial conditioning $(𝑋 2, \dots, 𝑋 𝑘) ∣ 𝑋 1 = 𝑥 1 \sim M u l t 𝑘 - 1 (𝑛 - 𝑛 1, (𝑝' 2, \dots, 𝑝' 𝑘))$ where $𝑝' 𝑗 = 𝑝 𝑗 𝑝 2 + \dots + 𝑝 𝑘$
$C o v [𝑋 𝑖, 𝑋 𝑗] = - 𝑛 𝑝 𝑖 𝑝 𝑗$ for $𝑖 \neq 𝑗$ .

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