Partition of variance into expected values

Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Variance ▷ Partition into expected values

Theorem: Let $X$ be a random variable. Then, the variance of $X$ is equal to the mean of the square of $X$ minus the square of the mean of $X$:

\[\label{eq:var-mean} \mathrm{Var}(X) = \mathrm{E}(X^2) - \mathrm{E}(X)^2 \; .\]

Proof: The variance of $X$ is defined as

\[\label{eq:var} \mathrm{Var}(X) = \mathrm{E}\left[ \left( X - \mathrm{E}[X] \right)^2 \right]\]

which, due to the linearity of the expected value, can be rewritten as

\[\label{eq:var-mean-qed} \begin{split} \mathrm{Var}(X) &= \mathrm{E}\left[ \left( X - \mathrm{E}[X] \right)^2 \right] \\ &= \mathrm{E}\left[ X^2 - 2 \, X \, \mathrm{E}(X) + \mathrm{E}(X)^2 \right] \\ &= \mathrm{E}(X^2) - 2 \, \mathrm{E}(X) \, \mathrm{E}(X) + \mathrm{E}(X)^2 \\ &= \mathrm{E}(X^2) - \mathrm{E}(X)^2 \; . \end{split}\]

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Sources:

Wikipedia (2020): "Variance"; in: Wikipedia, the free encyclopedia, retrieved on 2020-05-19; URL: https://en.wikipedia.org/wiki/Variance#Definition.

Metadata: ID: P104 | shortcut: var-mean | author: JoramSoch | date: 2020-05-19, 00:17.