Distribution of $(XY)^Z$ if $(X,Y,Z)$ is i.i.d. uniform on $[0,1]$











up vote
6
down vote

favorite












$X,Y$ and $Z$ are independent uniformly distributed on $[0,1]$



How is random variable $(XY)^Z$ distributed?



I had an idea to logarithm this and use convolution integral for the sum, but I'm not sure it's possible.






probability-theory probability-distributions







share|cite|improve this question
























  • Why do you need to know? Also, does $(XY)^{frac{1}{2}}$ mean $sqrt{XY}$ or $-sqrt{XY}$.
    – Dilip Sarwate
    Dec 18 '12 at 23:45















up vote
6
down vote

favorite












$X,Y$ and $Z$ are independent uniformly distributed on $[0,1]$



How is random variable $(XY)^Z$ distributed?



I had an idea to logarithm this and use convolution integral for the sum, but I'm not sure it's possible.






probability-theory probability-distributions







share|cite|improve this question
























  • Why do you need to know? Also, does $(XY)^{frac{1}{2}}$ mean $sqrt{XY}$ or $-sqrt{XY}$.
    – Dilip Sarwate
    Dec 18 '12 at 23:45













up vote
6
down vote

favorite









up vote
6
down vote

favorite











$X,Y$ and $Z$ are independent uniformly distributed on $[0,1]$



How is random variable $(XY)^Z$ distributed?



I had an idea to logarithm this and use convolution integral for the sum, but I'm not sure it's possible.






probability-theory probability-distributions







share|cite|improve this question















$X,Y$ and $Z$ are independent uniformly distributed on $[0,1]$



How is random variable $(XY)^Z$ distributed?



I had an idea to logarithm this and use convolution integral for the sum, but I'm not sure it's possible.






probability-theory probability-distributions




probability-theory probability-distributions






share|cite|improve this question















share|cite|improve this question













share|cite|improve this question




share|cite|improve this question








edited Jun 1 '17 at 21:08









Did

245k23218453




245k23218453










asked Dec 18 '12 at 23:31









Xxx

354110




354110












  • Why do you need to know? Also, does $(XY)^{frac{1}{2}}$ mean $sqrt{XY}$ or $-sqrt{XY}$.
    – Dilip Sarwate
    Dec 18 '12 at 23:45


















  • Why do you need to know? Also, does $(XY)^{frac{1}{2}}$ mean $sqrt{XY}$ or $-sqrt{XY}$.
    – Dilip Sarwate
    Dec 18 '12 at 23:45
















Why do you need to know? Also, does $(XY)^{frac{1}{2}}$ mean $sqrt{XY}$ or $-sqrt{XY}$.
– Dilip Sarwate
Dec 18 '12 at 23:45




Why do you need to know? Also, does $(XY)^{frac{1}{2}}$ mean $sqrt{XY}$ or $-sqrt{XY}$.
– Dilip Sarwate
Dec 18 '12 at 23:45










3 Answers
3






active

oldest

votes

















up vote
9
down vote













Hints:




  • The random variable $X$ is uniform on $(0,1)$ if and only if $-log X$ is exponential with parameter $1$.


  • If $U$ and $V$ are independent and exponential with parameter $1$, then $U+V$ is gamma distributed $(2,1)$, that is, with density $wmapsto wmathrm e^{-w}mathbf 1_{wgt0}$.


  • If $W$ is gamma distributed $(2,1)$ and $T$ is uniform on $(0,1)$ and independent of $W$, then $WU$ is exponential with parameter $1$.



Conclusion:




  • If $X$, $Y$ and $Z$ are independent and uniform on $(0,1)$, then $(XY)^Z$ is uniform on $(0,1)$.






share|cite|improve this answer





















  • +1: this is a nicer argument than my calculation... Even though they rely on fact which the OP might or might not have already proven.
    – Fabian
    Dec 19 '12 at 0:02




















up vote
5
down vote













Given the simplicity of the result there must be a nice short way to obtain it. However, I did not find one so I present the long and complicated calculation.



The distribution of the random variable $W=(XY)^Z$ is given by:
$$begin{align}P(wgeq W) &= int_0^1!dxint_0^1!dyint_0^1!dz, theta(w-(xy)^z)\
&= int_0^1!dxint_0^1!dy max{1-log_{xy} w,0}\
&=int_0^1!detaint_eta^1!frac{dx}{x}max{1-log_{eta} w,0} \
&=-int_0^w!deta log eta (1-log_{eta} w)\
&=w.
end{align}$$
with $eta=xy$.



Thus the variable $W$ is also uniformly distributed (between 0 and 1).






share|cite|improve this answer




























    up vote
    -1
    down vote













    Using the definition of weak convergence, it is so easy. First, for any positive integer k>=0,
    E{W^k}=1/(k+1)=EU^k. Hence, for any polynomial f(x), we have Ef(W)=Ef(U). For any bounded and continuous function g(.), we can find a polynomial function f(.) such that f can approximate g uniformly by Weierstrass's theorem. Thus, Eg(W)=Eg(U). So W~U(0, 1).






    share|cite|improve this answer





















    • You can format your answer using latex.
      – user60610
      Apr 18 '13 at 14:17










    • Hmmm... By the way, is this so obvious that E{W^k}=1/(k+1)?
      – Did
      Nov 27 at 19:00











    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',
    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%2f261783%2fdistribution-of-xyz-if-x-y-z-is-i-i-d-uniform-on-0-1%23new-answer', 'question_page');
    }
    );

    Post as a guest















    Required, but never shown

























    3 Answers
    3






    active

    oldest

    votes








    3 Answers
    3






    active

    oldest

    votes









    active

    oldest

    votes






    active

    oldest

    votes








    up vote
    9
    down vote













    Hints:




    • The random variable $X$ is uniform on $(0,1)$ if and only if $-log X$ is exponential with parameter $1$.


    • If $U$ and $V$ are independent and exponential with parameter $1$, then $U+V$ is gamma distributed $(2,1)$, that is, with density $wmapsto wmathrm e^{-w}mathbf 1_{wgt0}$.


    • If $W$ is gamma distributed $(2,1)$ and $T$ is uniform on $(0,1)$ and independent of $W$, then $WU$ is exponential with parameter $1$.



    Conclusion:




    • If $X$, $Y$ and $Z$ are independent and uniform on $(0,1)$, then $(XY)^Z$ is uniform on $(0,1)$.






    share|cite|improve this answer





















    • +1: this is a nicer argument than my calculation... Even though they rely on fact which the OP might or might not have already proven.
      – Fabian
      Dec 19 '12 at 0:02

















    up vote
    9
    down vote













    Hints:




    • The random variable $X$ is uniform on $(0,1)$ if and only if $-log X$ is exponential with parameter $1$.


    • If $U$ and $V$ are independent and exponential with parameter $1$, then $U+V$ is gamma distributed $(2,1)$, that is, with density $wmapsto wmathrm e^{-w}mathbf 1_{wgt0}$.


    • If $W$ is gamma distributed $(2,1)$ and $T$ is uniform on $(0,1)$ and independent of $W$, then $WU$ is exponential with parameter $1$.



    Conclusion:




    • If $X$, $Y$ and $Z$ are independent and uniform on $(0,1)$, then $(XY)^Z$ is uniform on $(0,1)$.






    share|cite|improve this answer





















    • +1: this is a nicer argument than my calculation... Even though they rely on fact which the OP might or might not have already proven.
      – Fabian
      Dec 19 '12 at 0:02















    up vote
    9
    down vote










    up vote
    9
    down vote









    Hints:




    • The random variable $X$ is uniform on $(0,1)$ if and only if $-log X$ is exponential with parameter $1$.


    • If $U$ and $V$ are independent and exponential with parameter $1$, then $U+V$ is gamma distributed $(2,1)$, that is, with density $wmapsto wmathrm e^{-w}mathbf 1_{wgt0}$.


    • If $W$ is gamma distributed $(2,1)$ and $T$ is uniform on $(0,1)$ and independent of $W$, then $WU$ is exponential with parameter $1$.



    Conclusion:




    • If $X$, $Y$ and $Z$ are independent and uniform on $(0,1)$, then $(XY)^Z$ is uniform on $(0,1)$.






    share|cite|improve this answer












    Hints:




    • The random variable $X$ is uniform on $(0,1)$ if and only if $-log X$ is exponential with parameter $1$.


    • If $U$ and $V$ are independent and exponential with parameter $1$, then $U+V$ is gamma distributed $(2,1)$, that is, with density $wmapsto wmathrm e^{-w}mathbf 1_{wgt0}$.


    • If $W$ is gamma distributed $(2,1)$ and $T$ is uniform on $(0,1)$ and independent of $W$, then $WU$ is exponential with parameter $1$.



    Conclusion:




    • If $X$, $Y$ and $Z$ are independent and uniform on $(0,1)$, then $(XY)^Z$ is uniform on $(0,1)$.







    share|cite|improve this answer












    share|cite|improve this answer



    share|cite|improve this answer










    answered Dec 19 '12 at 0:01









    Did

    245k23218453




    245k23218453












    • +1: this is a nicer argument than my calculation... Even though they rely on fact which the OP might or might not have already proven.
      – Fabian
      Dec 19 '12 at 0:02




















    • +1: this is a nicer argument than my calculation... Even though they rely on fact which the OP might or might not have already proven.
      – Fabian
      Dec 19 '12 at 0:02


















    +1: this is a nicer argument than my calculation... Even though they rely on fact which the OP might or might not have already proven.
    – Fabian
    Dec 19 '12 at 0:02






    +1: this is a nicer argument than my calculation... Even though they rely on fact which the OP might or might not have already proven.
    – Fabian
    Dec 19 '12 at 0:02












    up vote
    5
    down vote













    Given the simplicity of the result there must be a nice short way to obtain it. However, I did not find one so I present the long and complicated calculation.



    The distribution of the random variable $W=(XY)^Z$ is given by:
    $$begin{align}P(wgeq W) &= int_0^1!dxint_0^1!dyint_0^1!dz, theta(w-(xy)^z)\
    &= int_0^1!dxint_0^1!dy max{1-log_{xy} w,0}\
    &=int_0^1!detaint_eta^1!frac{dx}{x}max{1-log_{eta} w,0} \
    &=-int_0^w!deta log eta (1-log_{eta} w)\
    &=w.
    end{align}$$
    with $eta=xy$.



    Thus the variable $W$ is also uniformly distributed (between 0 and 1).






    share|cite|improve this answer

























      up vote
      5
      down vote













      Given the simplicity of the result there must be a nice short way to obtain it. However, I did not find one so I present the long and complicated calculation.



      The distribution of the random variable $W=(XY)^Z$ is given by:
      $$begin{align}P(wgeq W) &= int_0^1!dxint_0^1!dyint_0^1!dz, theta(w-(xy)^z)\
      &= int_0^1!dxint_0^1!dy max{1-log_{xy} w,0}\
      &=int_0^1!detaint_eta^1!frac{dx}{x}max{1-log_{eta} w,0} \
      &=-int_0^w!deta log eta (1-log_{eta} w)\
      &=w.
      end{align}$$
      with $eta=xy$.



      Thus the variable $W$ is also uniformly distributed (between 0 and 1).






      share|cite|improve this answer























        up vote
        5
        down vote










        up vote
        5
        down vote









        Given the simplicity of the result there must be a nice short way to obtain it. However, I did not find one so I present the long and complicated calculation.



        The distribution of the random variable $W=(XY)^Z$ is given by:
        $$begin{align}P(wgeq W) &= int_0^1!dxint_0^1!dyint_0^1!dz, theta(w-(xy)^z)\
        &= int_0^1!dxint_0^1!dy max{1-log_{xy} w,0}\
        &=int_0^1!detaint_eta^1!frac{dx}{x}max{1-log_{eta} w,0} \
        &=-int_0^w!deta log eta (1-log_{eta} w)\
        &=w.
        end{align}$$
        with $eta=xy$.



        Thus the variable $W$ is also uniformly distributed (between 0 and 1).






        share|cite|improve this answer












        Given the simplicity of the result there must be a nice short way to obtain it. However, I did not find one so I present the long and complicated calculation.



        The distribution of the random variable $W=(XY)^Z$ is given by:
        $$begin{align}P(wgeq W) &= int_0^1!dxint_0^1!dyint_0^1!dz, theta(w-(xy)^z)\
        &= int_0^1!dxint_0^1!dy max{1-log_{xy} w,0}\
        &=int_0^1!detaint_eta^1!frac{dx}{x}max{1-log_{eta} w,0} \
        &=-int_0^w!deta log eta (1-log_{eta} w)\
        &=w.
        end{align}$$
        with $eta=xy$.



        Thus the variable $W$ is also uniformly distributed (between 0 and 1).







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 19 '12 at 0:00









        Fabian

        19.2k3674




        19.2k3674






















            up vote
            -1
            down vote













            Using the definition of weak convergence, it is so easy. First, for any positive integer k>=0,
            E{W^k}=1/(k+1)=EU^k. Hence, for any polynomial f(x), we have Ef(W)=Ef(U). For any bounded and continuous function g(.), we can find a polynomial function f(.) such that f can approximate g uniformly by Weierstrass's theorem. Thus, Eg(W)=Eg(U). So W~U(0, 1).






            share|cite|improve this answer





















            • You can format your answer using latex.
              – user60610
              Apr 18 '13 at 14:17










            • Hmmm... By the way, is this so obvious that E{W^k}=1/(k+1)?
              – Did
              Nov 27 at 19:00















            up vote
            -1
            down vote













            Using the definition of weak convergence, it is so easy. First, for any positive integer k>=0,
            E{W^k}=1/(k+1)=EU^k. Hence, for any polynomial f(x), we have Ef(W)=Ef(U). For any bounded and continuous function g(.), we can find a polynomial function f(.) such that f can approximate g uniformly by Weierstrass's theorem. Thus, Eg(W)=Eg(U). So W~U(0, 1).






            share|cite|improve this answer





















            • You can format your answer using latex.
              – user60610
              Apr 18 '13 at 14:17










            • Hmmm... By the way, is this so obvious that E{W^k}=1/(k+1)?
              – Did
              Nov 27 at 19:00













            up vote
            -1
            down vote










            up vote
            -1
            down vote









            Using the definition of weak convergence, it is so easy. First, for any positive integer k>=0,
            E{W^k}=1/(k+1)=EU^k. Hence, for any polynomial f(x), we have Ef(W)=Ef(U). For any bounded and continuous function g(.), we can find a polynomial function f(.) such that f can approximate g uniformly by Weierstrass's theorem. Thus, Eg(W)=Eg(U). So W~U(0, 1).






            share|cite|improve this answer












            Using the definition of weak convergence, it is so easy. First, for any positive integer k>=0,
            E{W^k}=1/(k+1)=EU^k. Hence, for any polynomial f(x), we have Ef(W)=Ef(U). For any bounded and continuous function g(.), we can find a polynomial function f(.) such that f can approximate g uniformly by Weierstrass's theorem. Thus, Eg(W)=Eg(U). So W~U(0, 1).







            share|cite|improve this answer












            share|cite|improve this answer



            share|cite|improve this answer










            answered Apr 18 '13 at 13:57









            user73223

            1




            1












            • You can format your answer using latex.
              – user60610
              Apr 18 '13 at 14:17










            • Hmmm... By the way, is this so obvious that E{W^k}=1/(k+1)?
              – Did
              Nov 27 at 19:00


















            • You can format your answer using latex.
              – user60610
              Apr 18 '13 at 14:17










            • Hmmm... By the way, is this so obvious that E{W^k}=1/(k+1)?
              – Did
              Nov 27 at 19:00
















            You can format your answer using latex.
            – user60610
            Apr 18 '13 at 14:17




            You can format your answer using latex.
            – user60610
            Apr 18 '13 at 14:17












            Hmmm... By the way, is this so obvious that E{W^k}=1/(k+1)?
            – Did
            Nov 27 at 19:00




            Hmmm... By the way, is this so obvious that E{W^k}=1/(k+1)?
            – Did
            Nov 27 at 19:00


















            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.





            Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


            Please pay close attention to the following guidance:


            • 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.


            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%2f261783%2fdistribution-of-xyz-if-x-y-z-is-i-i-d-uniform-on-0-1%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

            Berounka

            Image
            Berounka.mw-parser-output .entete.map{background-image:url("//upload.wikimedia.org/wikipedia/commons/7/7a/Picto_infobox_map.png")} Caractéristiques Longueur 139  km Bassin 8 861  km 2 Bassin collecteur Elbe Débit moyen ? Cours Source Confluence de la Mže et de la Radbuza · Localisation Pilsen · Altitude 305  m · Coordonnées 49° 45′ 13″ N, 13° 23′ 24″ E Confluence Vltava · Localisation Prague · Altitude 187  m · Coordonnées 49° 59′ 43″ N, 14° 24′ 04″ E Géographie Pays traversés   République tchèque modifier   La Berounka est une rivière de la République tchèque. Elle porte le nom de Mže depuis sa source en Allemagne près de la frontière jusqu'à sa confluence avec la Radbuza à Plzeň. Elle continue ensuite sous le nom de Berounka jusqu'à sa confluence avec la Vltava, près de Prague. La rivière est une destination prisée pour les amateurs de canoë, qui appréc...
            Read more

            Different font size/position of beamer's navigation symbols template's content depending on regular/plain...

            Image
            3 Let's suppose I'd like to divert the navigation symbols template (e.g. in order replace them by the frame numbers, see this bug report ;) I would be facing troubles regarding the font size/position, depending on regular/plain frames, as shown by the following MCE. How could I get rid of these troubles? documentclass{beamer} usetheme{PaloAlto} setbeamertemplate{navigation symbols}{insertframenumber/inserttotalframenumber} begin{document} begin{frame} frametitle{A regular frame} end{frame} begin{frame}[plain] frametitle{A plain frame} end{frame} end{document} beamer header-footer share | improve this question asked Dec 10 '18 a...
            Read more

            Sphinx de Gizeh

            Image
            Pour les articles homonymes, voir Sphinx. Sphinx de Gizeh.mw-parser-output .entete.egypte-antique{background-image:url("//upload.wikimedia.org/wikipedia/commons/c/cc/Picto_infobox_ancient_Egypt.png")} Site d'Égypte antique Le visage du sphinx de Gizeh. Noms en arabe أبو الهول , Abou al-Hôl Localisation Région Basse-Égypte Coordonnées 29° 58′ 31″ nord, 31° 08′ 16″ est Géolocalisation sur la carte : Égypte Sphinx de Gizeh modifier   Le sphinx de Gizeh est la statue thérianthrope qui se dresse devant les grandes pyramides du plateau de Gizeh, en Basse-Égypte. Sculpture monumentale monolithique la plus grande du monde [ 1 ] avec 73,5 mètres de longueur, 14 mètres de largeur et 20,22 mètres de hauteur, elle représente un sphinx couchant. Réalisée vers -2500, elle serait attribuée à Djédefrê, l'un des pharaons de la IV e  dynastie, représentant son père le pharaon Khéops. Sommaire 1 Étymologie 2 Desc...
            Read more