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.

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

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

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    $\begingroup$ Thank you. But is there any other reason why the one is use than the other? $\endgroup$
    – isemaj
    Jan 14 '19 at 11:42
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    $\begingroup$ @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

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 $\langle X-\mu, X-\mu\rangle$


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