Proof that Variance is Non-Negative


\[\textbf{Var}(x) = \mathbb{E}[ (X - \mathbb{E}(X) )^2 ] = \mathbb{E}(X^2) - \mathbb{E}(X)^2\]

Show that variance is never negative.

There’s 2 ways to do this: the right way and overkill.

The Right Way

Notice that you’re taking the mean of a real-valued function squared, which is never negative. Therefore, the mean can’t be negative.

The Way That It Actually Went Down

Jensen’s Inequality.

The square function is convex. Therefore, by Jensen’s inequality, the positive term in the last expression dominates the negative one, proving the statement.

This is way overkill for something that can be figured out without even writing something down.

Hindsight is rather insulting.

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