What does the angle bracket mean in variance formula?
$begingroup$
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.
variance notation
edited Jan 14 at 11:22
Nick Cox
38.4k483128
asked Jan 14 at 9:29
isemajisemaj
313
$endgroup$
add a comment |
$begingroup$
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.
variance notation
edited Jan 14 at 11:22
Nick Cox
38.4k483128
asked Jan 14 at 9:29
isemajisemaj
313
$endgroup$
1
$begingroup$
math world defines it: and <X> denotes the expectation value of X.
$endgroup$
– seanv507
Jan 14 at 10:00
1
$begingroup$
see mathworld.wolfram.com/AngleBracket.html - the last sentence of the article proper.
$endgroup$
– Glen_b♦
Jan 14 at 13:04
1
$begingroup$
It means a physicist (or possibly a pure mathematician) is writing about probability :-).
$endgroup$
– whuber♦
Jan 14 at 15:44
add a comment |
$begingroup$
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.
variance notation
edited Jan 14 at 11:22
Nick Cox
38.4k483128
asked Jan 14 at 9:29
isemajisemaj
313
$endgroup$
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.
variance notation
variance notation
edited Jan 14 at 11:22
Nick Cox
38.4k483128
asked Jan 14 at 9:29
isemajisemaj
313
edited Jan 14 at 11:22
Nick Cox
38.4k483128
asked Jan 14 at 9:29
isemajisemaj
313
edited Jan 14 at 11:22
Nick Cox
38.4k483128
edited Jan 14 at 11:22
Nick Cox
38.4k483128
edited Jan 14 at 11:22
Nick Cox
38.4k483128
38.4k483128
asked Jan 14 at 9:29
isemajisemaj
313
asked Jan 14 at 9:29
isemajisemaj
313
asked Jan 14 at 9:29
isemajisemaj
313
313
1
$begingroup$
math world defines it: and <X> denotes the expectation value of X.
$endgroup$
– seanv507
Jan 14 at 10:00
1
$begingroup$
see mathworld.wolfram.com/AngleBracket.html - the last sentence of the article proper.
$endgroup$
– Glen_b♦
Jan 14 at 13:04
1
$begingroup$
It means a physicist (or possibly a pure mathematician) is writing about probability :-).
$endgroup$
– whuber♦
Jan 14 at 15:44
add a comment |
1
$begingroup$
math world defines it: and <X> denotes the expectation value of X.
$endgroup$
– seanv507
Jan 14 at 10:00
1
$begingroup$
see mathworld.wolfram.com/AngleBracket.html - the last sentence of the article proper.
$endgroup$
– Glen_b♦
Jan 14 at 13:04
1
$begingroup$
It means a physicist (or possibly a pure mathematician) is writing about probability :-).
$endgroup$
– whuber♦
Jan 14 at 15:44
1
1
$begingroup$
math world defines it: and <X> denotes the expectation value of X.
$endgroup$
– seanv507
Jan 14 at 10:00
$begingroup$
math world defines it: and <X> denotes the expectation value of X.
$endgroup$
– seanv507
Jan 14 at 10:00
1
1
$begingroup$
see mathworld.wolfram.com/AngleBracket.html - the last sentence of the article proper.
$endgroup$
– Glen_b♦
Jan 14 at 13:04
$begingroup$
see mathworld.wolfram.com/AngleBracket.html - the last sentence of the article proper.
$endgroup$
– Glen_b♦
Jan 14 at 13:04
1
1
$begingroup$
It means a physicist (or possibly a pure mathematician) is writing about probability :-).
$endgroup$
– whuber♦
Jan 14 at 15:44
$begingroup$
It means a physicist (or possibly a pure mathematician) is writing about probability :-).
$endgroup$
– whuber♦
Jan 14 at 15:44
add a comment |
2 Answers
2
active
oldest
votes
$begingroup$
It's the expected value of $(X-mu)^2$, i.e., it's the same as $sigma^2=E[(X-mu)^2]$.
answered Jan 14 at 9:38
FrederikDSFrederikDS
612
$endgroup$
1
$begingroup$
Thank you. But is there any other reason why the one is use than the other?
$endgroup$
– isemaj
Jan 14 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 at 20:14
add a comment |
$begingroup$
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 $langle X-mu, X-murangle$
answered Jan 14 at 9:34
OmGOmG
31628
$endgroup$
add a comment |
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
It's the expected value of $(X-mu)^2$, i.e., it's the same as $sigma^2=E[(X-mu)^2]$.
answered Jan 14 at 9:38
FrederikDSFrederikDS
612
$endgroup$
1
$begingroup$
Thank you. But is there any other reason why the one is use than the other?
$endgroup$
– isemaj
Jan 14 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 at 20:14
add a comment |
$begingroup$
It's the expected value of $(X-mu)^2$, i.e., it's the same as $sigma^2=E[(X-mu)^2]$.
answered Jan 14 at 9:38
FrederikDSFrederikDS
612
$endgroup$
1
$begingroup$
Thank you. But is there any other reason why the one is use than the other?
$endgroup$
– isemaj
Jan 14 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 at 20:14
add a comment |
$begingroup$
It's the expected value of $(X-mu)^2$, i.e., it's the same as $sigma^2=E[(X-mu)^2]$.
answered Jan 14 at 9:38
FrederikDSFrederikDS
612
$endgroup$
It's the expected value of $(X-mu)^2$, i.e., it's the same as $sigma^2=E[(X-mu)^2]$.
answered Jan 14 at 9:38
FrederikDSFrederikDS
612
answered Jan 14 at 9:38
FrederikDSFrederikDS
612
answered Jan 14 at 9:38
FrederikDSFrederikDS
612
answered Jan 14 at 9:38
FrederikDSFrederikDS
612
612
1
$begingroup$
Thank you. But is there any other reason why the one is use than the other?
$endgroup$
– isemaj
Jan 14 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 at 20:14
add a comment |
1
$begingroup$
Thank you. But is there any other reason why the one is use than the other?
$endgroup$
– isemaj
Jan 14 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 at 20:14
1
1
$begingroup$
Thank you. But is there any other reason why the one is use than the other?
$endgroup$
– isemaj
Jan 14 at 11:42
$begingroup$
Thank you. But is there any other reason why the one is use than the other?
$endgroup$
– isemaj
Jan 14 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 at 20:14
$begingroup$
@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 at 20:14
add a comment |
$begingroup$
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 $langle X-mu, X-murangle$
answered Jan 14 at 9:34
OmGOmG
31628
$endgroup$
add a comment |
$begingroup$
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 $langle X-mu, X-murangle$
answered Jan 14 at 9:34
OmGOmG
31628
$endgroup$
add a comment |
$begingroup$
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 $langle X-mu, X-murangle$
answered Jan 14 at 9:34
OmGOmG
31628
$endgroup$
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 $langle X-mu, X-murangle$
answered Jan 14 at 9:34
OmGOmG
31628
answered Jan 14 at 9:34
OmGOmG
31628
answered Jan 14 at 9:34
OmGOmG
31628
answered Jan 14 at 9:34
OmGOmG
31628
31628
add a comment |
add a comment |
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1
$begingroup$
math world defines it: and <X> denotes the expectation value of X.
$endgroup$
– seanv507
Jan 14 at 10:00
1
$begingroup$
see mathworld.wolfram.com/AngleBracket.html - the last sentence of the article proper.
$endgroup$
– Glen_b♦
Jan 14 at 13:04
1
$begingroup$
It means a physicist (or possibly a pure mathematician) is writing about probability :-).
$endgroup$
– whuber♦
Jan 14 at 15:44