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?

  • 3
    $\begingroup$ 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. $\endgroup$
    – Glen_b
    Commented Oct 24, 2018 at 21:10
  • 1
    $\begingroup$ @Glen_b I couldn't see that, thanks. THIS is an explicit demonstration of what you've said I believe. $\endgroup$
    – HBat
    Commented Oct 24, 2018 at 21:25
  • $\begingroup$ 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. $\endgroup$
    – Glen_b
    Commented Oct 24, 2018 at 21:41


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