What is the purpose of the constant in the probability density function The 2019 Stack Overflow Developer Survey Results Are InConfusion between probability distribution function and probability density functionProbability density function vs. probability mass functionBound 1D gaussian domain in the interval $[-3sigma, 3sigma]$ so it still is a probability density functionCan a probability density function be used directly as probability function?Probability density of a function of a random variableis this function increasing or decreasing on what intervals?Homework: questions about probability distribution functions and probability density functionGaussian function constantDeriving the Covariance of Multivariate Gaussianprobability density function of a function of a random variable?

What is this 4-propeller plane?

Should I use my personal or workplace e-mail when registering to external websites for work purpose?

Inflated grade on resume at previous job, might former employer tell new employer?

If the Wish spell is used to duplicate the effect of Simulacrum, are existing duplicates destroyed?

What can other administrators access on my machine?

Why isn't airport relocation done gradually?

How can I fix this gap between bookcases I made?

Does a dangling wire really electrocute me if I'm standing in water?

How long do I have to send payment?

How can I create a character who can assume the widest possible range of creature sizes?

Why can Shazam do this?

Is there a name of the flying bionic bird?

How are circuits which use complex ICs normally simulated?

CiviEvent: Public link for events of a specific type

What does Linus Torvalds mean when he says that Git "never ever" tracks a file?

I see my dog run

In microwave frequencies, do you use a circulator when you need a (near) perfect diode?

Which Sci-Fi work first showed weapon of galactic-scale mass destruction?

How to reverse every other sublist of a list?

Patience, young "Padovan"

How come people say “Would of”?

Does it makes sense to buy a new cycle to learn riding?

What is the best strategy for white in this position?

Lethal sonic weapons



What is the purpose of the constant in the probability density function



The 2019 Stack Overflow Developer Survey Results Are InConfusion between probability distribution function and probability density functionProbability density function vs. probability mass functionBound 1D gaussian domain in the interval $[-3sigma, 3sigma]$ so it still is a probability density functionCan a probability density function be used directly as probability function?Probability density of a function of a random variableis this function increasing or decreasing on what intervals?Homework: questions about probability distribution functions and probability density functionGaussian function constantDeriving the Covariance of Multivariate Gaussianprobability density function of a function of a random variable?










1












$begingroup$


I have been studying the probability density function...



$$frac1sigma sqrt2 pie^frac(-(x - mu ))^22sigma ^2$$



For now I remove the constant, and using the following proof, I prove that...



$$int_-infty^inftye^frac-x^22 = sqrt2 pi $$



The way I interpret this is that the area under the gaussian distribution is $sqrt2 pi $. But I am still having a hard time figuring out what the constant is doing. It seems to divide by the area itself and by $sigma$ as well. Why is this done?






probability statistics probability-distributions gaussian-integral







share|cite|improve this question









$endgroup$







  • 1




    $begingroup$
    so the integral of the probability density function over the entire space is equal to one
    $endgroup$
    – J. W. Tanner
    3 hours ago
















1












$begingroup$


I have been studying the probability density function...



$$frac1sigma sqrt2 pie^frac(-(x - mu ))^22sigma ^2$$



For now I remove the constant, and using the following proof, I prove that...



$$int_-infty^inftye^frac-x^22 = sqrt2 pi $$



The way I interpret this is that the area under the gaussian distribution is $sqrt2 pi $. But I am still having a hard time figuring out what the constant is doing. It seems to divide by the area itself and by $sigma$ as well. Why is this done?






probability statistics probability-distributions gaussian-integral







share|cite|improve this question









$endgroup$







  • 1




    $begingroup$
    so the integral of the probability density function over the entire space is equal to one
    $endgroup$
    – J. W. Tanner
    3 hours ago














1












1








1





$begingroup$


I have been studying the probability density function...



$$frac1sigma sqrt2 pie^frac(-(x - mu ))^22sigma ^2$$



For now I remove the constant, and using the following proof, I prove that...



$$int_-infty^inftye^frac-x^22 = sqrt2 pi $$



The way I interpret this is that the area under the gaussian distribution is $sqrt2 pi $. But I am still having a hard time figuring out what the constant is doing. It seems to divide by the area itself and by $sigma$ as well. Why is this done?






probability statistics probability-distributions gaussian-integral







share|cite|improve this question









$endgroup$




I have been studying the probability density function...



$$frac1sigma sqrt2 pie^frac(-(x - mu ))^22sigma ^2$$



For now I remove the constant, and using the following proof, I prove that...



$$int_-infty^inftye^frac-x^22 = sqrt2 pi $$



The way I interpret this is that the area under the gaussian distribution is $sqrt2 pi $. But I am still having a hard time figuring out what the constant is doing. It seems to divide by the area itself and by $sigma$ as well. Why is this done?






probability statistics probability-distributions gaussian-integral




probability statistics probability-distributions gaussian-integral






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 3 hours ago









BolboaBolboa

398516




398516







  • 1




    $begingroup$
    so the integral of the probability density function over the entire space is equal to one
    $endgroup$
    – J. W. Tanner
    3 hours ago













  • 1




    $begingroup$
    so the integral of the probability density function over the entire space is equal to one
    $endgroup$
    – J. W. Tanner
    3 hours ago








1




1




$begingroup$
so the integral of the probability density function over the entire space is equal to one
$endgroup$
– J. W. Tanner
3 hours ago





$begingroup$
so the integral of the probability density function over the entire space is equal to one
$endgroup$
– J. W. Tanner
3 hours ago











2 Answers
2






active

oldest

votes


















2












$begingroup$

If you consider every possible outcome of some event you should expect the probability of it happening to be $1$, not $sqrt2pi$ so the constant scales the distribution to conform with the normal convention of ascribing a probability between zero and one.






share|cite|improve this answer









$endgroup$








  • 1




    $begingroup$
    (In case its not clear, which it probably is) More precisely, you want the sum of the probabilities of all possible events to equal $1$, and since an integral represents a sum over such continuous variables, that's what this is.
    $endgroup$
    – John Doe
    3 hours ago



















2












$begingroup$

It is doing that, but observe that you are also stretching in the horizontal direction by the same factor (in the exponential). Say if $sigma>1$ you are decreasing your area by a factor $sigma$ but you are increasing it by the same factor because you replace $x$ by $x/sigma$ (the shift does not change the area of course)






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: "69"
    ;
    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: true,
    noModals: true,
    showLowRepImageUploadWarning: true,
    reputationToPostImages: 10,
    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
    ,
    noCode: true, onDemand: true,
    discardSelector: ".discard-answer"
    ,immediatelyShowMarkdownHelp:true
    );



    );













    draft saved

    draft discarded


















    StackExchange.ready(
    function ()
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3181774%2fwhat-is-the-purpose-of-the-constant-in-the-probability-density-function%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









    2












    $begingroup$

    If you consider every possible outcome of some event you should expect the probability of it happening to be $1$, not $sqrt2pi$ so the constant scales the distribution to conform with the normal convention of ascribing a probability between zero and one.






    share|cite|improve this answer









    $endgroup$








    • 1




      $begingroup$
      (In case its not clear, which it probably is) More precisely, you want the sum of the probabilities of all possible events to equal $1$, and since an integral represents a sum over such continuous variables, that's what this is.
      $endgroup$
      – John Doe
      3 hours ago
















    2












    $begingroup$

    If you consider every possible outcome of some event you should expect the probability of it happening to be $1$, not $sqrt2pi$ so the constant scales the distribution to conform with the normal convention of ascribing a probability between zero and one.






    share|cite|improve this answer









    $endgroup$








    • 1




      $begingroup$
      (In case its not clear, which it probably is) More precisely, you want the sum of the probabilities of all possible events to equal $1$, and since an integral represents a sum over such continuous variables, that's what this is.
      $endgroup$
      – John Doe
      3 hours ago














    2












    2








    2





    $begingroup$

    If you consider every possible outcome of some event you should expect the probability of it happening to be $1$, not $sqrt2pi$ so the constant scales the distribution to conform with the normal convention of ascribing a probability between zero and one.






    share|cite|improve this answer









    $endgroup$



    If you consider every possible outcome of some event you should expect the probability of it happening to be $1$, not $sqrt2pi$ so the constant scales the distribution to conform with the normal convention of ascribing a probability between zero and one.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered 3 hours ago









    CyclotomicFieldCyclotomicField

    2,4931314




    2,4931314







    • 1




      $begingroup$
      (In case its not clear, which it probably is) More precisely, you want the sum of the probabilities of all possible events to equal $1$, and since an integral represents a sum over such continuous variables, that's what this is.
      $endgroup$
      – John Doe
      3 hours ago













    • 1




      $begingroup$
      (In case its not clear, which it probably is) More precisely, you want the sum of the probabilities of all possible events to equal $1$, and since an integral represents a sum over such continuous variables, that's what this is.
      $endgroup$
      – John Doe
      3 hours ago








    1




    1




    $begingroup$
    (In case its not clear, which it probably is) More precisely, you want the sum of the probabilities of all possible events to equal $1$, and since an integral represents a sum over such continuous variables, that's what this is.
    $endgroup$
    – John Doe
    3 hours ago





    $begingroup$
    (In case its not clear, which it probably is) More precisely, you want the sum of the probabilities of all possible events to equal $1$, and since an integral represents a sum over such continuous variables, that's what this is.
    $endgroup$
    – John Doe
    3 hours ago












    2












    $begingroup$

    It is doing that, but observe that you are also stretching in the horizontal direction by the same factor (in the exponential). Say if $sigma>1$ you are decreasing your area by a factor $sigma$ but you are increasing it by the same factor because you replace $x$ by $x/sigma$ (the shift does not change the area of course)






    share|cite|improve this answer









    $endgroup$

















      2












      $begingroup$

      It is doing that, but observe that you are also stretching in the horizontal direction by the same factor (in the exponential). Say if $sigma>1$ you are decreasing your area by a factor $sigma$ but you are increasing it by the same factor because you replace $x$ by $x/sigma$ (the shift does not change the area of course)






      share|cite|improve this answer









      $endgroup$















        2












        2








        2





        $begingroup$

        It is doing that, but observe that you are also stretching in the horizontal direction by the same factor (in the exponential). Say if $sigma>1$ you are decreasing your area by a factor $sigma$ but you are increasing it by the same factor because you replace $x$ by $x/sigma$ (the shift does not change the area of course)






        share|cite|improve this answer









        $endgroup$



        It is doing that, but observe that you are also stretching in the horizontal direction by the same factor (in the exponential). Say if $sigma>1$ you are decreasing your area by a factor $sigma$ but you are increasing it by the same factor because you replace $x$ by $x/sigma$ (the shift does not change the area of course)







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 3 hours ago









        GReyesGReyes

        2,39815




        2,39815



























            draft saved

            draft discarded
















































            Thanks for contributing an answer to Mathematics Stack Exchange!


            • 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%2fmath.stackexchange.com%2fquestions%2f3181774%2fwhat-is-the-purpose-of-the-constant-in-the-probability-density-function%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

            名間水力發電廠 目录 沿革 設施 鄰近設施 註釋 外部連結 导航菜单23°50′10″N 120°42′41″E / 23.83611°N 120.71139°E / 23.83611; 120.7113923°50′10″N 120°42′41″E / 23.83611°N 120.71139°E / 23.83611; 120.71139計畫概要原始内容臺灣第一座BOT 模式開發的水力發電廠－名間水力電廠名間水力發電廠 水利署首件BOT案原始内容《小檔案》名間電廠 首座BOT水力發電廠原始内容名間電廠BOT - 經濟部水利署中區水資源局编编

            Image
            東螺溪清水溪加走寮溪阿里山溪東埔蚋溪獅尾堀清水溪清水溝溪水里溪陳有蘭溪郡坑溪十八重溪那馬嘎班溪松柏坑溪卓棍溪瓠瓢坑溪益則坑溪玉崙溪丹大溪郡大溪巒大溪卡社溪栗栖溪萬大溪萬大北溪萬大南溪霧社溪塔羅灣溪馬赫坡溪萬大發電廠大觀發電廠日月潭第一發電所明潭發電廠...
            Read more

            Prove that NP is closed under karp reduction?Space(n) not closed under Karp reductions - what about NTime(n)?Class P is closed under rotation?Prove or disprove that $NL$ is closed under polynomial many-one reductions$mathbfNC_2$ is closed under log-space reductionOn Karp reductionwhen can I know if a class (complexity) is closed under reduction (cook/karp)Check if class $PSPACE$ is closed under polyonomially space reductionIs NPSPACE also closed under polynomial-time reduction and under log-space reduction?Prove PSPACE is closed under complement?Prove PSPACE is closed under union?

            Image
            Is a conference paper whose proceedings will be published in IEEE Xplore counted as a publication? Prove that NP is closed under karp reduction? Is it important to consider tone, melody, and musical form while writing a song? How is it possible to have an ability score that is less than 3? Can I make popcorn with any corn? Mathematical cryptic clues tikz: show 0 at the axis origin Minkowski space can i play a electric guitar through a bass amp? Why not use SQL instead of GraphQL? How can I prevent hyper evolved versions of regular creatures from wiping out their cousins? How old can references or sources in a thesis be? Do VLANs within a subnet need to have their own subnet for router on a stick? How to add double frame in tcolorbox? How much RAM could one put in a typical 80386 setup? "You are your self first supporter", a more proper way to say it How is the claim "I am in New York only if I am in America" the same as "If I am ...
            Read more

            Is my guitar’s action too high? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)Strings too stiff on a recently purchased acoustic guitar | Cort AD880CEIs the action of my guitar really high?Μy little finger is too weak to play guitarWith guitar, how long should I give my fingers to strengthen / callous?When playing a fret the guitar sounds mutedPlaying (Barre) chords up the guitar neckI think my guitar strings are wound too tight and I can't play barre chordsF barre chord on an SG guitarHow to find to the right strings of a barre chord by feel?High action on higher fret on my steel acoustic guitar

            Image
            Is the Mordenkainen's Sword spell underpowered? Escaping $ in listings environment Short story about astronauts fertilizing soil with their own bodies No invitation for tourist visa but I want to visit Do regular languages belong to Space(1)? Can I take recommendation from someone I met at a conference? How to name indistinguishable henchmen in a screenplay? Found this skink in my tomato plant bucket. Is he trapped? Or could he leave if he wanted? How can I prevent/balance waiting and turtling as a response to cooldown mechanics Can I feed enough spin up electron to a black hole to affect it's angular momentum? Why does BitLocker not use RSA? Is my guitar’s action too high? Does GDPR cover the collection of data by websites that crawl the web and resell user data Trying to enter the Fox's den How to achieve cat-like agility? "Destructive power" carried by a B-52? Should man-made satellites feature an intelligent inverted "cow c...
            Read more