What does the angle bracket mean in variance formula?












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.






variance notation







share|cite|improve this question











$endgroup$








  • 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


















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.






variance notation







share|cite|improve this question











$endgroup$








  • 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
















6












6








6


1



$begingroup$


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







share|cite|improve this question











$endgroup$




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






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jan 14 at 11:22









Nick Cox

38.4k483128




38.4k483128










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
















  • 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












2 Answers
2






active

oldest

votes


















6












$begingroup$

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






share|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 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



















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 $langle X-mu, X-murangle$






share|cite|improve this answer









$endgroup$













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    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    6












    $begingroup$

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






    share|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 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
















    6












    $begingroup$

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






    share|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 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














    6












    6








    6





    $begingroup$

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






    share|cite|improve this answer









    $endgroup$



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







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    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














    • 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













    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 $langle X-mu, X-murangle$






    share|cite|improve this answer









    $endgroup$


















      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 $langle X-mu, X-murangle$






      share|cite|improve this answer









      $endgroup$
















        4












        4








        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 $langle X-mu, X-murangle$






        share|cite|improve this answer









        $endgroup$



        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 $langle X-mu, X-murangle$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Jan 14 at 9:34









        OmGOmG

        31628




        31628






























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