Index: The Book of Statistical Proofs ▷ Probability Distributions ▷ Multivariate discrete distributions ▷ Multinomial distribution ▷ Mean

Theorem: Let $X$ be a random vector following a multinomial distribution:

\[\label{eq:mult} X \sim \mathrm{Mult}(n,\left[p_1, \ldots, p_k \right]) \; .\]

Then, the mean or expected value of $X$ is

\[\label{eq:bin-mean} \mathrm{E}(X) = \left[n p_1, \ldots, n p_k \right] \; .\]

Proof: By definition, a multinomial random variable is the sum of $n$ independent and identical categorical trials with category probabilities $p_1, \ldots, p_k$. Therefore, the expected value is

\[\label{eq:mult-mean-s1} \mathrm{E}(X) = \mathrm{E}(X_1 + \ldots + X_n)\]

and because the expected value is a linear operator, this is equal to

\[\label{eq:mult-mean-s2} \begin{split} \mathrm{E}(X) &= \mathrm{E}(X_1) + \ldots + \mathrm{E}(X_n) \\ &= \sum_{i=1}^{n} \mathrm{E}(X_i) \; . \end{split}\]

With the expected value of the categorical distribution, we have:

\[\label{eq:mult-mean-s3} \mathrm{E}(X) = \sum_{i=1}^{n} \left[p_1, \ldots, p_k \right] = n \cdot \left[p_1, \ldots, p_k \right] = \left[n p_1, \ldots, n p_k \right] \; .\]
∎
Sources:

Metadata: ID: P25 | shortcut: mult-mean | author: JoramSoch | date: 2020-01-16, 11:26.