Index: The Book of Statistical ProofsProbability 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:

$\label{eq:mult-mean-s3} \mathrm{E}(X) = \sum_{i=1}^{n} \left[p_1, \ldots, p_k \right] = n \cdot \left[p_1, \ldots, p_k \right] = \left[n p_1, \ldots, n p_k \right] \; .$
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Metadata: ID: P25 | shortcut: mult-mean | author: JoramSoch | date: 2020-01-16, 11:26.