Proof: Non-negativity of the variance
Index: The Book of Statistical Proofs ▷ General Theorems ▷ Probability theory ▷ Variance ▷ Non-negativity
Metadata: ID: P123 | shortcut: var-nonneg | author: JoramSoch | date: 2020-06-06, 07:29.
Theorem: The variance is always non-negative, i.e.
\[\label{eq:var-nonneg} \mathrm{Var}(X) \geq 0 \; .\]Proof: The variance of a random variable is defined as
\[\label{eq:var} \mathrm{Var}(X) = \mathrm{E}\left[ (X-\mathrm{E}(X))^2 \right] \; .\]
1) If $X$ is a discrete random variable, then, because squares and probabilities are strictly non-negative, all the addends in
are also non-negative, thus the entire sum must be non-negative.
2) If $X$ is a continuous random variable, then, because squares and probability densities are strictly non-negative, the integrand in
is always non-negative, thus the term on the right-hand side is a Lebesgue integral, so that the result on the left-hand side must be non-negative.
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Sources: - Wikipedia (2020): "Variance"; in: Wikipedia, the free encyclopedia, retrieved on 2020-06-06; URL: https://en.wikipedia.org/wiki/Variance#Basic_properties.
Metadata: ID: P123 | shortcut: var-nonneg | author: JoramSoch | date: 2020-06-06, 07:29.