Covariance of Random Variable with Itself

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

$\blacksquare$