# Why is the sum of two variances is larger than two times the covariances?

I saw the following inequality:

$$\sigma_{X_1}^2 + \sigma_{X_2}^2 \geq 2\sigma_{X_1X_2}$$

and the brief explanation said that it is based on Cauchy-Schwarz inequality. But I couldn't make the connection. Why is the equation above is true?

• Well, I'd have said it's implied by $\text{Var}(X_1-X_2)\geq 0$ but it sounds like that's not intended to be used here. Going from Cauchy Schwarz should be straightforward though; write the relevant expectations for each term in your equation and identify appropriate vectors $u$ and $v$ in those, and apply the inequality. Commented Oct 24, 2018 at 21:10
• @Glen_b I couldn't see that, thanks. THIS is an explicit demonstration of what you've said I believe.
– HBat
Commented Oct 24, 2018 at 21:25
• If you apply Cauchy Schwarz, directly, you could consider whether writing $u$ as an expectation involving $X_1$ (and so on for other terms) would get you there. Commented Oct 24, 2018 at 21:41