what is the log of the PDF for a Normal Distribution? Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)How to solve/compute for normal distribution and log-normal CDF inverse?Distribution of the convolution of squared normal and chi-squared variables?Cramer's theorem for a precise normal asymptotic distributionConditional Expected Value of Product of Normal and Log-Normal DistributionAsymptotic relation for a class of probability distribution functionsShow that $Y_1+Y_2$ have distribution skew-normalExpected Fisher's information matrix for Student's t-distribution?Expected Value of Maximum likelihood mean for Gaussian DistributionJoint density of the sum of a random and a non-random variable?Reversing conditional distribution

Special flights

Relating to the President and obstruction, were Mueller's conclusions preordained?

Why complex landing gears are used instead of simple,reliability and light weight muscle wire or shape memory alloys?

License to disallow distribution in closed source software, but allow exceptions made by owner?

What are the main differences between Stargate SG-1 cuts?

NERDTreeMenu Remapping

What would you call this weird metallic apparatus that allows you to lift people?

Flight departed from the gate 5 min before scheduled departure time. Refund options

Is there hard evidence that the grant peer review system performs significantly better than random?

How can a team of shapeshifters communicate?

Where is the Next Backup Size entry on iOS 12?

Putting class ranking in CV, but against dept guidelines

If Windows 7 doesn't support WSL, then what is "Subsystem for UNIX-based Applications"?

Trying to understand entropy as a novice in thermodynamics

My mentor says to set image to Fine instead of RAW — how is this different from JPG?

Is it dangerous to install hacking tools on my private linux machine?

"klopfte jemand" or "jemand klopfte"?

What does it mean that physics no longer uses mechanical models to describe phenomena?

Is multiple magic items in one inherently imbalanced?

Found this skink in my tomato plant bucket. Is he trapped? Or could he leave if he wanted?

Did Mueller's report provide an evidentiary basis for the claim of Russian govt election interference via social media?

Moving a wrapfig vertically to encroach partially on a subsection title

Why is std::move not [[nodiscard]] in C++20?

What is the difference between CTSS and ITS?



what is the log of the PDF for a Normal Distribution?



Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern)How to solve/compute for normal distribution and log-normal CDF inverse?Distribution of the convolution of squared normal and chi-squared variables?Cramer's theorem for a precise normal asymptotic distributionConditional Expected Value of Product of Normal and Log-Normal DistributionAsymptotic relation for a class of probability distribution functionsShow that $Y_1+Y_2$ have distribution skew-normalExpected Fisher's information matrix for Student's t-distribution?Expected Value of Maximum likelihood mean for Gaussian DistributionJoint density of the sum of a random and a non-random variable?Reversing conditional distribution



.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;








1












$begingroup$


I am learning Maximum Likelihood Estimation.



per this post, the log of the PDF for a Normal Distribution looks like this.



enter image description here



let's call this equation1.



according to any probability theory textbook the formula of the PDF for a Normal Distribution:



$$
frac 1sigma sqrt 2pi
e^-frac (x - mu)^22sigma ^2
,-infty <x<infty
$$



taking log produces:



beginalign
ln(frac 1sigma sqrt 2pi
e^-frac (x - mu)^22sigma ^2) &=
ln(frac 1sigma sqrt 2pi)+ln(e^-frac (x - mu)^22sigma ^2)\
&=-ln(sigma)-frac12 ln(2pi) - frac (x - mu)^22sigma ^2
endalign



which is very different from equation1.



is equation1 right? what am I missing?










share|cite|improve this question









$endgroup$







  • 3




    $begingroup$
    Your first equation is the joint log-pdf of a sample of n iid normal random variables (AKA the log-likelihood of that sample). The second equation is the the log-pdf of a single normal random variable
    $endgroup$
    – Artem Mavrin
    2 hours ago











  • $begingroup$
    @ArtemMavrin, I think your comment would be a perfectly good answer if you expanded on just a bit to make it slightly more clear.
    $endgroup$
    – StatsStudent
    2 hours ago

















1












$begingroup$


I am learning Maximum Likelihood Estimation.



per this post, the log of the PDF for a Normal Distribution looks like this.



enter image description here



let's call this equation1.



according to any probability theory textbook the formula of the PDF for a Normal Distribution:



$$
frac 1sigma sqrt 2pi
e^-frac (x - mu)^22sigma ^2
,-infty <x<infty
$$



taking log produces:



beginalign
ln(frac 1sigma sqrt 2pi
e^-frac (x - mu)^22sigma ^2) &=
ln(frac 1sigma sqrt 2pi)+ln(e^-frac (x - mu)^22sigma ^2)\
&=-ln(sigma)-frac12 ln(2pi) - frac (x - mu)^22sigma ^2
endalign



which is very different from equation1.



is equation1 right? what am I missing?










share|cite|improve this question









$endgroup$







  • 3




    $begingroup$
    Your first equation is the joint log-pdf of a sample of n iid normal random variables (AKA the log-likelihood of that sample). The second equation is the the log-pdf of a single normal random variable
    $endgroup$
    – Artem Mavrin
    2 hours ago











  • $begingroup$
    @ArtemMavrin, I think your comment would be a perfectly good answer if you expanded on just a bit to make it slightly more clear.
    $endgroup$
    – StatsStudent
    2 hours ago













1












1








1





$begingroup$


I am learning Maximum Likelihood Estimation.



per this post, the log of the PDF for a Normal Distribution looks like this.



enter image description here



let's call this equation1.



according to any probability theory textbook the formula of the PDF for a Normal Distribution:



$$
frac 1sigma sqrt 2pi
e^-frac (x - mu)^22sigma ^2
,-infty <x<infty
$$



taking log produces:



beginalign
ln(frac 1sigma sqrt 2pi
e^-frac (x - mu)^22sigma ^2) &=
ln(frac 1sigma sqrt 2pi)+ln(e^-frac (x - mu)^22sigma ^2)\
&=-ln(sigma)-frac12 ln(2pi) - frac (x - mu)^22sigma ^2
endalign



which is very different from equation1.



is equation1 right? what am I missing?










share|cite|improve this question









$endgroup$




I am learning Maximum Likelihood Estimation.



per this post, the log of the PDF for a Normal Distribution looks like this.



enter image description here



let's call this equation1.



according to any probability theory textbook the formula of the PDF for a Normal Distribution:



$$
frac 1sigma sqrt 2pi
e^-frac (x - mu)^22sigma ^2
,-infty <x<infty
$$



taking log produces:



beginalign
ln(frac 1sigma sqrt 2pi
e^-frac (x - mu)^22sigma ^2) &=
ln(frac 1sigma sqrt 2pi)+ln(e^-frac (x - mu)^22sigma ^2)\
&=-ln(sigma)-frac12 ln(2pi) - frac (x - mu)^22sigma ^2
endalign



which is very different from equation1.



is equation1 right? what am I missing?







probability log






share|cite|improve this question













share|cite|improve this question











share|cite|improve this question




share|cite|improve this question










asked 2 hours ago









shi95shi95

103




103







  • 3




    $begingroup$
    Your first equation is the joint log-pdf of a sample of n iid normal random variables (AKA the log-likelihood of that sample). The second equation is the the log-pdf of a single normal random variable
    $endgroup$
    – Artem Mavrin
    2 hours ago











  • $begingroup$
    @ArtemMavrin, I think your comment would be a perfectly good answer if you expanded on just a bit to make it slightly more clear.
    $endgroup$
    – StatsStudent
    2 hours ago












  • 3




    $begingroup$
    Your first equation is the joint log-pdf of a sample of n iid normal random variables (AKA the log-likelihood of that sample). The second equation is the the log-pdf of a single normal random variable
    $endgroup$
    – Artem Mavrin
    2 hours ago











  • $begingroup$
    @ArtemMavrin, I think your comment would be a perfectly good answer if you expanded on just a bit to make it slightly more clear.
    $endgroup$
    – StatsStudent
    2 hours ago







3




3




$begingroup$
Your first equation is the joint log-pdf of a sample of n iid normal random variables (AKA the log-likelihood of that sample). The second equation is the the log-pdf of a single normal random variable
$endgroup$
– Artem Mavrin
2 hours ago





$begingroup$
Your first equation is the joint log-pdf of a sample of n iid normal random variables (AKA the log-likelihood of that sample). The second equation is the the log-pdf of a single normal random variable
$endgroup$
– Artem Mavrin
2 hours ago













$begingroup$
@ArtemMavrin, I think your comment would be a perfectly good answer if you expanded on just a bit to make it slightly more clear.
$endgroup$
– StatsStudent
2 hours ago




$begingroup$
@ArtemMavrin, I think your comment would be a perfectly good answer if you expanded on just a bit to make it slightly more clear.
$endgroup$
– StatsStudent
2 hours ago










1 Answer
1






active

oldest

votes


















2












$begingroup$

For a single observed value $x$ you have log-likelihood:



$$ell_x(mu,sigma^2) = - ln sigma - frac12 ln (2 pi) - frac12 Big( fracx-musigma Big)^2.$$



For a sample of observed values $mathbfx = (x_1,...,x_n)$ you then have:



$$ell_mathbfx(mu,sigma^2) = sum_i=1^n ell_x(mu,sigma^2) = - n ln sigma - fracn2 ln (2 pi) - frac12 sigma^2 sum_i=1^n (x_i-mu)^2.$$






share|cite|improve this answer









$endgroup$













    Your Answer








    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
    );



    );













    draft saved

    draft discarded


















    StackExchange.ready(
    function ()
    StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fstats.stackexchange.com%2fquestions%2f404191%2fwhat-is-the-log-of-the-pdf-for-a-normal-distribution%23new-answer', 'question_page');

    );

    Post as a guest















    Required, but never shown

























    1 Answer
    1






    active

    oldest

    votes








    1 Answer
    1






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes









    2












    $begingroup$

    For a single observed value $x$ you have log-likelihood:



    $$ell_x(mu,sigma^2) = - ln sigma - frac12 ln (2 pi) - frac12 Big( fracx-musigma Big)^2.$$



    For a sample of observed values $mathbfx = (x_1,...,x_n)$ you then have:



    $$ell_mathbfx(mu,sigma^2) = sum_i=1^n ell_x(mu,sigma^2) = - n ln sigma - fracn2 ln (2 pi) - frac12 sigma^2 sum_i=1^n (x_i-mu)^2.$$






    share|cite|improve this answer









    $endgroup$

















      2












      $begingroup$

      For a single observed value $x$ you have log-likelihood:



      $$ell_x(mu,sigma^2) = - ln sigma - frac12 ln (2 pi) - frac12 Big( fracx-musigma Big)^2.$$



      For a sample of observed values $mathbfx = (x_1,...,x_n)$ you then have:



      $$ell_mathbfx(mu,sigma^2) = sum_i=1^n ell_x(mu,sigma^2) = - n ln sigma - fracn2 ln (2 pi) - frac12 sigma^2 sum_i=1^n (x_i-mu)^2.$$






      share|cite|improve this answer









      $endgroup$















        2












        2








        2





        $begingroup$

        For a single observed value $x$ you have log-likelihood:



        $$ell_x(mu,sigma^2) = - ln sigma - frac12 ln (2 pi) - frac12 Big( fracx-musigma Big)^2.$$



        For a sample of observed values $mathbfx = (x_1,...,x_n)$ you then have:



        $$ell_mathbfx(mu,sigma^2) = sum_i=1^n ell_x(mu,sigma^2) = - n ln sigma - fracn2 ln (2 pi) - frac12 sigma^2 sum_i=1^n (x_i-mu)^2.$$






        share|cite|improve this answer









        $endgroup$



        For a single observed value $x$ you have log-likelihood:



        $$ell_x(mu,sigma^2) = - ln sigma - frac12 ln (2 pi) - frac12 Big( fracx-musigma Big)^2.$$



        For a sample of observed values $mathbfx = (x_1,...,x_n)$ you then have:



        $$ell_mathbfx(mu,sigma^2) = sum_i=1^n ell_x(mu,sigma^2) = - n ln sigma - fracn2 ln (2 pi) - frac12 sigma^2 sum_i=1^n (x_i-mu)^2.$$







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered 1 hour ago









        BenBen

        28.9k233129




        28.9k233129



























            draft saved

            draft discarded
















































            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%2f404191%2fwhat-is-the-log-of-the-pdf-for-a-normal-distribution%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

            Are there any AGPL-style licences that require source code modifications to be public? Planned maintenance scheduled April 23, 2019 at 23:30 UTC (7:30pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?Force derivative works to be publicAre there any GPL like licenses for Apple App Store?Do you violate the GPL if you provide source code that cannot be compiled?GPL - is it distribution to use libraries in an appliance loaned to customers?Distributing App for free which uses GPL'ed codeModifications of server software under GPL, with web/CLI interfaceDoes using an AGPLv3-licensed library prevent me from dual-licensing my own source code?Can I publish only select code under GPLv3 from a private project?Is there published precedent regarding the scope of covered work that uses AGPL software?If MIT licensed code links to GPL licensed code what should be the license of the resulting binary program?If I use a public API endpoint that has its source code licensed under AGPL in my app, do I need to disclose my source?

            2013 GY136 Descoberta | Órbita | Referências Menu de navegação«List Of Centaurs and Scattered-Disk Objects»«List of Known Trans-Neptunian Objects»

            Mortes em março de 2019 Referências Menu de navegação«Zhores Alferov, Nobel de Física bielorrusso, morre aos 88 anos - Ciência»«Fallece Rafael Torija, o bispo emérito de Ciudad Real»«Peter Hurford dies at 88»«Keith Flint, vocalista do The Prodigy, morre aos 49 anos»«Luke Perry, ator de 'Barrados no baile' e 'Riverdale', morre aos 52 anos»«Former Rangers and Scotland captain Eric Caldow dies, aged 84»«Morreu, aos 61 anos, a antiga lenda do wrestling King Kong Bundy»«Fallece el actor y director teatral Abraham Stavans»«In Memoriam Guillaume Faye»«Sidney Sheinberg, a Force Behind Universal and Spielberg, Is Dead at 84»«Carmine Persico, Colombo Crime Family Boss, Is Dead at 85»«Dirigent Michael Gielen gestorben»«Ciclista tricampeã mundial e prata na Rio 2016 é encontrada morta em casa aos 23 anos»«Pagan Community Notes: Raven Grimassi dies, Indianapolis pop-up event cancelled, Circle Sanctuary announces new podcast, and more!»«Hal Blaine, Wrecking Crew Drummer, Dies at 90»«Morre Coutinho, que editou dupla lendária com Pelé no Santos»«Cantor Demétrius, ídolo da Jovem Guarda, morre em SP»«Ex-presidente do Vasco, Eurico Miranda morre no Rio de Janeiro»«Bronze no Mundial de basquete de 1971, Laís Elena morre aos 76 anos»«Diretor de Corridas da F1, Charlie Whiting morre aos 66 anos às vésperas do GP da Austrália»«Morreu o cardeal Danneels, da Bélgica»«Morreu o cartoonista Augusto Cid»«Morreu a atriz Maria Isabel de Lizandra, de "Vale Tudo" e novelas da Tupi»«WS Merwin, prize-winning poet of nature, dies at 91»«Atriz Márcia Real morre em São Paulo aos 88 anos»«Mauritanie: décès de l'ancien président Mohamed Mahmoud ould Louly»«Morreu Dick Dale, o rei da surf guitar e de "Pulp Fiction"»«Falleció Víctor Genes»«João Carlos Marinho, autor de 'O Gênio do Crime', morre em SP»«Legendary Horror Director and SFX Artist John Carl Buechler Dies at 66»«Morre em Salvador a religiosa Makota Valdina»«مرگ بازیکن‌ سابق نساجی بر اثر سقوط سنگ در مازندران»«Domingos Oliveira morre no Rio»«Morre Airton Ravagniani, ex-São Paulo, Fla, Vasco, Grêmio e Sport - Notícias»«Morre o escritor Flavio Moreira da Costa»«Larry Cohen, Writer-Director of 'It's Alive' and 'Hell Up in Harlem,' Dies at 77»«Scott Walker, experimental singer-songwriter, dead at 76»«Joseph Pilato, Day of the Dead Star and Horror Favorite, Dies at 70»«Sheffield United set to pay tribute to legendary goalkeeper Ted Burgin who has died at 91»«Morre Rafael Henzel, sobrevivente de acidente aéreo da Chapecoense»«Morre Valery Bykovsky, um dos primeiros cosmonautas da União Soviética»«Agnès Varda, cineasta da Nouvelle Vague, morre aos 90 anos»«Agnès Varda, cineasta francesa, morre aos 90 anos»«Tania Mallet, James Bond Actress and Helen Mirren's Cousin, Dies at 77»e