2
$\begingroup$

Suppose I have two finite sets of data A and B, with equal length n.

What's the best upper and lower bound I can make on var(A+B), in terms of var(A) and var(B)?

$\endgroup$
4
  • 2
    $\begingroup$ This question simplifies greatly when recast in terms of the variances, due to the simple formula for var(A+B). That version is addressed by the Cauchy-Schwarz Inequality. $\endgroup$
    – whuber
    Oct 3, 2011 at 17:32
  • $\begingroup$ Okay, I've now found $var(A+B) \leq var(A) + var(B) + 2 \sqrt{var(A) var(B)}$, but is this bound tight? $\endgroup$ Oct 3, 2011 at 17:39
  • $\begingroup$ @Greg Snow's example (where $A=B$) demonstrates tightness. $\endgroup$
    – whuber
    Oct 3, 2011 at 18:46
  • $\begingroup$ The upper bound can be written a bit more cleanly in terms of standard deviations $sd(A + B) \le sd(A) + sd(B)$. $\endgroup$
    – passerby51
    Jan 27, 2023 at 21:06

1 Answer 1

2
$\begingroup$

The lower bound will occur when $B = -A$ so that $A + B = 0$ and the variance is $0$. This also occurs when $B$ is shifted from $A$, but in any case you cannot have a variance less than $0$ and this shows a case where it can equal $0$.

The upper bound comes when $B = A$ or $B = c1 + c2 \times A$ which gives a correlation of $1$. In this case you get the variance is the comments above.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.