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









New contributor




isemaj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$








  • 1




    $begingroup$
    math world defines it: and <X> denotes the expectation value of X.
    $endgroup$
    – seanv507
    18 hours ago






  • 1




    $begingroup$
    see mathworld.wolfram.com/AngleBracket.html - the last sentence of the article proper.
    $endgroup$
    – Glen_b♦
    15 hours ago






  • 1




    $begingroup$
    It means a physicist (or possibly a pure mathematician) is writing about probability :-).
    $endgroup$
    – whuber♦
    13 hours ago


















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









New contributor




isemaj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$








  • 1




    $begingroup$
    math world defines it: and <X> denotes the expectation value of X.
    $endgroup$
    – seanv507
    18 hours ago






  • 1




    $begingroup$
    see mathworld.wolfram.com/AngleBracket.html - the last sentence of the article proper.
    $endgroup$
    – Glen_b♦
    15 hours ago






  • 1




    $begingroup$
    It means a physicist (or possibly a pure mathematician) is writing about probability :-).
    $endgroup$
    – whuber♦
    13 hours ago
















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









New contributor




isemaj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







$endgroup$




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









New contributor




isemaj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.











share|cite|improve this question









New contributor




isemaj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









share|cite|improve this question




share|cite|improve this question








edited 17 hours ago









Nick Cox

38.2k483128




38.2k483128






New contributor




isemaj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.









asked 19 hours ago









isemajisemaj

312




312




New contributor




isemaj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.





New contributor





isemaj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






isemaj is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.








  • 1




    $begingroup$
    math world defines it: and <X> denotes the expectation value of X.
    $endgroup$
    – seanv507
    18 hours ago






  • 1




    $begingroup$
    see mathworld.wolfram.com/AngleBracket.html - the last sentence of the article proper.
    $endgroup$
    – Glen_b♦
    15 hours ago






  • 1




    $begingroup$
    It means a physicist (or possibly a pure mathematician) is writing about probability :-).
    $endgroup$
    – whuber♦
    13 hours ago
















  • 1




    $begingroup$
    math world defines it: and <X> denotes the expectation value of X.
    $endgroup$
    – seanv507
    18 hours ago






  • 1




    $begingroup$
    see mathworld.wolfram.com/AngleBracket.html - the last sentence of the article proper.
    $endgroup$
    – Glen_b♦
    15 hours ago






  • 1




    $begingroup$
    It means a physicist (or possibly a pure mathematician) is writing about probability :-).
    $endgroup$
    – whuber♦
    13 hours ago










1




1




$begingroup$
math world defines it: and <X> denotes the expectation value of X.
$endgroup$
– seanv507
18 hours ago




$begingroup$
math world defines it: and <X> denotes the expectation value of X.
$endgroup$
– seanv507
18 hours ago




1




1




$begingroup$
see mathworld.wolfram.com/AngleBracket.html - the last sentence of the article proper.
$endgroup$
– Glen_b♦
15 hours ago




$begingroup$
see mathworld.wolfram.com/AngleBracket.html - the last sentence of the article proper.
$endgroup$
– Glen_b♦
15 hours ago




1




1




$begingroup$
It means a physicist (or possibly a pure mathematician) is writing about probability :-).
$endgroup$
– whuber♦
13 hours ago






$begingroup$
It means a physicist (or possibly a pure mathematician) is writing about probability :-).
$endgroup$
– whuber♦
13 hours ago












2 Answers
2






active

oldest

votes


















5












$begingroup$

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








New contributor




FrederikDS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.






$endgroup$









  • 1




    $begingroup$
    Thank you. But is there any other reason why the one is use than the other?
    $endgroup$
    – isemaj
    17 hours ago










  • $begingroup$
    @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.
    8 hours ago



















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









$endgroup$













    Your Answer





    StackExchange.ifUsing("editor", function () {
    return StackExchange.using("mathjaxEditing", function () {
    StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
    StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
    });
    });
    }, "mathjax-editing");

    StackExchange.ready(function() {
    var channelOptions = {
    tags: "".split(" "),
    id: "65"
    };
    initTagRenderer("".split(" "), "".split(" "), channelOptions);

    StackExchange.using("externalEditor", function() {
    // Have to fire editor after snippets, if snippets enabled
    if (StackExchange.settings.snippets.snippetsEnabled) {
    StackExchange.using("snippets", function() {
    createEditor();
    });
    }
    else {
    createEditor();
    }
    });

    function createEditor() {
    StackExchange.prepareEditor({
    heartbeatType: 'answer',
    autoActivateHeartbeat: false,
    convertImagesToLinks: false,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: null,
    bindNavPrevention: true,
    postfix: "",
    imageUploader: {
    brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
    contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
    allowUrls: true
    },
    onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    });


    }
    });






    isemaj is a new contributor. Be nice, and check out our Code of Conduct.










    draft saved

    draft discarded


















    StackExchange.ready(
    function () {
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fstats.stackexchange.com%2fquestions%2f387078%2fwhat-does-the-angle-bracket-mean-in-variance-formula%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    2 Answers
    2






    active

    oldest

    votes








    2 Answers
    2






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    5












    $begingroup$

    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








    New contributor




    FrederikDS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






    $endgroup$









    • 1




      $begingroup$
      Thank you. But is there any other reason why the one is use than the other?
      $endgroup$
      – isemaj
      17 hours ago










    • $begingroup$
      @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.
      8 hours ago
















    5












    $begingroup$

    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








    New contributor




    FrederikDS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






    $endgroup$









    • 1




      $begingroup$
      Thank you. But is there any other reason why the one is use than the other?
      $endgroup$
      – isemaj
      17 hours ago










    • $begingroup$
      @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.
      8 hours ago














    5












    5








    5





    $begingroup$

    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








    New contributor




    FrederikDS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






    $endgroup$



    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








    New contributor




    FrederikDS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.









    share|cite|improve this answer



    share|cite|improve this answer






    New contributor




    FrederikDS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.









    answered 19 hours ago









    FrederikDSFrederikDS

    512




    512




    New contributor




    FrederikDS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.





    New contributor





    FrederikDS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.






    FrederikDS is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.








    • 1




      $begingroup$
      Thank you. But is there any other reason why the one is use than the other?
      $endgroup$
      – isemaj
      17 hours ago










    • $begingroup$
      @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.
      8 hours ago














    • 1




      $begingroup$
      Thank you. But is there any other reason why the one is use than the other?
      $endgroup$
      – isemaj
      17 hours ago










    • $begingroup$
      @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.
      8 hours ago








    1




    1




    $begingroup$
    Thank you. But is there any other reason why the one is use than the other?
    $endgroup$
    – isemaj
    17 hours ago




    $begingroup$
    Thank you. But is there any other reason why the one is use than the other?
    $endgroup$
    – isemaj
    17 hours ago












    $begingroup$
    @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.
    8 hours ago




    $begingroup$
    @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.
    8 hours ago













    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 19 hours ago









        OmGOmG

        31628




        31628






















            isemaj is a new contributor. Be nice, and check out our Code of Conduct.










            draft saved

            draft discarded


















            isemaj is a new contributor. Be nice, and check out our Code of Conduct.













            isemaj is a new contributor. Be nice, and check out our Code of Conduct.












            isemaj is a new contributor. Be nice, and check out our Code of Conduct.
















            Thanks for contributing an answer to Cross Validated!


            • Please be sure to answer the question. Provide details and share your research!

            But avoid …



            • Asking for help, clarification, or responding to other answers.

            • Making statements based on opinion; back them up with references or personal experience.


            Use MathJax to format equations. MathJax reference.


            To learn more, see our tips on writing great answers.




            draft saved


            draft discarded














            StackExchange.ready(
            function () {
            StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fstats.stackexchange.com%2fquestions%2f387078%2fwhat-does-the-angle-bracket-mean-in-variance-formula%23new-answer', 'question_page');
            }
            );

            Post as a guest















            Required, but never shown





















































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown

































            Required, but never shown














            Required, but never shown












            Required, but never shown







            Required, but never shown







            -

            Popular posts from this blog

            Сан-Квентин

            Image
            Сан-Квентин Местоположение Сан-Квентин, Калифорния Координаты 37°56′13″ с. ш. 122°29′20″ з. д. H G Я O L Текущий статус Действует Режим безопасности минимальный и максимальный Количество мест 3 302 Открытие Июнь 1852 Находится в ведомстве California Department of Corrections and Rehabilitation Начальник Mike Martel, начальник  Сан-Квентин на Викискладе Сан-Квентин (англ.  San Quentin ) — тюрьма, располагающаяся на 432 акрах (около 1,7 км²), на мысе Сан-Квентин, в округе Марин, штат Калифорния, США. Тюрьма штата Калифорния Сан-Квентин была открыта в июле 1852 года и является старейшей в штате. Она была построена заключёнными, которые проживали в течение строительства на тюремном судне Вабан. В Сан-Квентине отбывали наказание и мужчины, и женщины до 1934 года, когда построили женскую тюрьму в Техачапи и туда перевели всех заключённых женщин. В Сан-Квентине приводятся в исполнение смертные казни, для этого имеется газовая камера, но...
            Read more

            8-я гвардейская общевойсковая армия

            Image
            8-я гвардейская общевойсковая ордена Ленина армия Награды: Войска: РККА → Советская армия → СВ России Род войск: Формирование: 1943 Расформирование (преобразование): 1992 Предшественник: 62-я армия (1942—43) → 8-я гвардейская общевойсковая армия (1943—92) → 8-й гвардейский армейский корпус (1992—98) Боевой путь Изюм-Барвенковская операция Донбасская операция Запорожская операция Битва за Днепр Днепропетровская операция Березнеговато-Снигирёвская операция Одесская операция Люблин-Брестская операция Варшавско-Познанская операция Восточно-Померанская операция Берлинская наступательная операция У этого термина существуют и другие значения, см. 8-я армия. Фотография Гвардейского Красного Знамени (лицевая сторона) 8-й гвардейской армии , образца 1943 года. Фотография Гвардейского Красного Знамени (оборотная сторона) 8-й гвардейской армии , образца 1943 года. 8-я гвардейская общевойсковая ордена Ленина армия  — гвардейское фор...
            Read more

            Алькесар

            Image
            Муниципалитет Алькесар Alquézar Герб 42°10′16″ с. ш. 0°01′27″ в. д. H G Я O L Страна Испания   Испания Автономное сообщество Арагон Провинция Уэска Комарка Сомонтано-де-Барбастро Алькальд Хосе Марияно Альтемир Ласкорс (ИСРП) История и география Площадь 32,36 км² Высота НУМ 660 м Часовой пояс UTC+1, летом UTC+2 Население Население 313 человек ( 2010 ) Плотность 9,92 чел./км² Этнохороним alquezarano/-a Цифровые идентификаторы Телефонный код +34 974 Почтовые индексы 22112 Автомобильный код HU alquezar.org   (исп.) Показать/скрыть карты Алькесар Алькесар Алькесар  Аудио, фото и видео на Викискладе Алькесар (исп.  Alquézar ) — муниципалитет в Испании, входит в провинцию Уэска, в составе автономного сообщества Арагон. Муниципалитет находится в составе района (комарки) Сомонтано-де-Барбастро. Занимает площадь 32,36 км². Население — 321 челове...
            Read more