6
$\begingroup$

When I check the formula of variance in Mathworld which is

$$ \sigma^2 \equiv \langle\ (X - \mu)^2 \rangle\ $$

Though I'm more familiar with the other formula, I just wanted to know what does the angle bracket mean aside from the formula in variance.

$\endgroup$
  • 1
    $\begingroup$ math world defines it: and <X> denotes the expectation value of X. $\endgroup$ – seanv507 Jan 14 '19 at 10:00
  • 1
    $\begingroup$ see mathworld.wolfram.com/AngleBracket.html - the last sentence of the article proper. $\endgroup$ – Glen_b Jan 14 '19 at 13:04
  • 1
    $\begingroup$ It means a physicist (or possibly a pure mathematician) is writing about probability :-). $\endgroup$ – whuber Jan 14 '19 at 15:44
7
$\begingroup$

It's the expected value of $(X-\mu)^2$, i.e., it's the same as $\sigma^2=E[(X-\mu)^2]$.

| cite | improve this answer | |
$\endgroup$
  • 1
    $\begingroup$ Thank you. But is there any other reason why the one is use than the other? $\endgroup$ – isemaj Jan 14 '19 at 11:42
  • $\begingroup$ @isemaj Besides OmG's answer, you might also be interested in the generalisation of expectations to matrix elements in the bra-ket formalism of quantum mechanics, which upon suppression of explicit states gives the "angle" formalism for expectations. $\endgroup$ – J.G. Jan 14 '19 at 20:14
4
$\begingroup$

It means an inner product for the multi-dimensional case. When $X \in \mathbb{R}^n$ and $n \geq 2$ and want to define variance, the definition of the variance is related to the inner product of $X-\mu$ to itself, and denoted as $\langle X-\mu, X-\mu\rangle$

| cite | improve this answer | |
$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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