Covariance of Random Variable with Itself
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Theorem
Let $X$ be a random variable.
Then $\cov {X, X} = \var X$.
Proof
We have:
\(\ds \cov {X, X}\) | \(=\) | \(\ds \expect {\paren {X - \expect X} \paren {X - \expect X} }\) | Definition of Covariance | |||||||||||
\(\ds \) | \(=\) | \(\ds \expect {\paren {X - \expect X}^2}\) | ||||||||||||
\(\ds \) | \(=\) | \(\ds \var X\) | Definition of Variance |
$\blacksquare$