If the expectation and variance of x are both not affected by y, and vice versa, then must x and y be...












1












$begingroup$


I know that if $mathbb{E}[X]=mathbb{E}[X|Y] , mathbb{E}[Y]=mathbb{E}[Y|X]$, $X$ and $Y$ can be dependent, for example a ‘uniform’ distribution in a unit circle.
Now we add the variance, if
$$mathbb{E}[X]=mathbb{E}[X|Y], mathbb{E}[Y]=mathbb{E}[Y|X], $$$$Var(X)=Var(X|Y), Var(Y)=Var(Y|X).$$
Say the expectation and variance of $X$ are both not affected by $Y$, and vice versa, then must $X$ and $Y$ be independent?
In this case I can not find a counterexample just like the uniform circle.



If they are independent, how to prove it? If not, is there a counterexample?



Thanks!






probability conditional-expectation independence







share|cite|improve this question









New contributor




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







$endgroup$

















    1












    $begingroup$


    I know that if $mathbb{E}[X]=mathbb{E}[X|Y] , mathbb{E}[Y]=mathbb{E}[Y|X]$, $X$ and $Y$ can be dependent, for example a ‘uniform’ distribution in a unit circle.
    Now we add the variance, if
    $$mathbb{E}[X]=mathbb{E}[X|Y], mathbb{E}[Y]=mathbb{E}[Y|X], $$$$Var(X)=Var(X|Y), Var(Y)=Var(Y|X).$$
    Say the expectation and variance of $X$ are both not affected by $Y$, and vice versa, then must $X$ and $Y$ be independent?
    In this case I can not find a counterexample just like the uniform circle.



    If they are independent, how to prove it? If not, is there a counterexample?



    Thanks!






    probability conditional-expectation independence







    share|cite|improve this question









    New contributor




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







    $endgroup$















      1












      1








      1





      $begingroup$


      I know that if $mathbb{E}[X]=mathbb{E}[X|Y] , mathbb{E}[Y]=mathbb{E}[Y|X]$, $X$ and $Y$ can be dependent, for example a ‘uniform’ distribution in a unit circle.
      Now we add the variance, if
      $$mathbb{E}[X]=mathbb{E}[X|Y], mathbb{E}[Y]=mathbb{E}[Y|X], $$$$Var(X)=Var(X|Y), Var(Y)=Var(Y|X).$$
      Say the expectation and variance of $X$ are both not affected by $Y$, and vice versa, then must $X$ and $Y$ be independent?
      In this case I can not find a counterexample just like the uniform circle.



      If they are independent, how to prove it? If not, is there a counterexample?



      Thanks!






      probability conditional-expectation independence







      share|cite|improve this question









      New contributor




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







      $endgroup$




      I know that if $mathbb{E}[X]=mathbb{E}[X|Y] , mathbb{E}[Y]=mathbb{E}[Y|X]$, $X$ and $Y$ can be dependent, for example a ‘uniform’ distribution in a unit circle.
      Now we add the variance, if
      $$mathbb{E}[X]=mathbb{E}[X|Y], mathbb{E}[Y]=mathbb{E}[Y|X], $$$$Var(X)=Var(X|Y), Var(Y)=Var(Y|X).$$
      Say the expectation and variance of $X$ are both not affected by $Y$, and vice versa, then must $X$ and $Y$ be independent?
      In this case I can not find a counterexample just like the uniform circle.



      If they are independent, how to prove it? If not, is there a counterexample?



      Thanks!






      probability conditional-expectation independence




      probability conditional-expectation independence






      share|cite|improve this question









      New contributor




      Brent Wagner 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




      Brent Wagner 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 2 hours ago







      Brent Wagner













      New contributor




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









      asked 4 hours ago









      Brent WagnerBrent Wagner

      133




      133




      New contributor




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





      New contributor





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






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






















          2 Answers
          2






          active

          oldest

          votes


















          3












          $begingroup$

          Consider $Xsim N(0,1)$ and $Ysim N(0,2)$ if $X>0$ and $Ysim t_4$ otherwise. Then $X, Y$ are dependent, but your conditions hold (the $t_4$ distribution has mean 0 and variance 2).






          share|cite|improve this answer









          $endgroup$





















            4












            $begingroup$

            Here's a discrete example:
            $$100P(x,y)=begin{array}{c|ccccc}xbackslash y&-2&-1&0&1&2\ hline -2&1&0&6&0&1\-1&0&9&0&9&0\0&6&0&36&0&6\1&0&9&0&9&0\2&1&0&6&0&1end{array}$$
            Condition on any particular $x$, and $y$ has mean zero and variance $1$. Condition on any particular $y$, and $x$ has mean zero and variance $1$. But, of course, they're not independent, as the distribution's support isn't a Cartesian product.






            share|cite|improve this answer









            $endgroup$













            • $begingroup$
              Thank you very much! But I can only adapt one answer so I adapted the earlier one... Your discrete example is wonderful!
              $endgroup$
              – Brent Wagner
              2 hours ago











            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
            });


            }
            });






            Brent Wagner 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%2fmath.stackexchange.com%2fquestions%2f3086758%2fif-the-expectation-and-variance-of-x-are-both-not-affected-by-y-and-vice-versa%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









            3












            $begingroup$

            Consider $Xsim N(0,1)$ and $Ysim N(0,2)$ if $X>0$ and $Ysim t_4$ otherwise. Then $X, Y$ are dependent, but your conditions hold (the $t_4$ distribution has mean 0 and variance 2).






            share|cite|improve this answer









            $endgroup$


















              3












              $begingroup$

              Consider $Xsim N(0,1)$ and $Ysim N(0,2)$ if $X>0$ and $Ysim t_4$ otherwise. Then $X, Y$ are dependent, but your conditions hold (the $t_4$ distribution has mean 0 and variance 2).






              share|cite|improve this answer









              $endgroup$
















                3












                3








                3





                $begingroup$

                Consider $Xsim N(0,1)$ and $Ysim N(0,2)$ if $X>0$ and $Ysim t_4$ otherwise. Then $X, Y$ are dependent, but your conditions hold (the $t_4$ distribution has mean 0 and variance 2).






                share|cite|improve this answer









                $endgroup$



                Consider $Xsim N(0,1)$ and $Ysim N(0,2)$ if $X>0$ and $Ysim t_4$ otherwise. Then $X, Y$ are dependent, but your conditions hold (the $t_4$ distribution has mean 0 and variance 2).







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered 3 hours ago









                DashermanDasherman

                1,065817




                1,065817























                    4












                    $begingroup$

                    Here's a discrete example:
                    $$100P(x,y)=begin{array}{c|ccccc}xbackslash y&-2&-1&0&1&2\ hline -2&1&0&6&0&1\-1&0&9&0&9&0\0&6&0&36&0&6\1&0&9&0&9&0\2&1&0&6&0&1end{array}$$
                    Condition on any particular $x$, and $y$ has mean zero and variance $1$. Condition on any particular $y$, and $x$ has mean zero and variance $1$. But, of course, they're not independent, as the distribution's support isn't a Cartesian product.






                    share|cite|improve this answer









                    $endgroup$













                    • $begingroup$
                      Thank you very much! But I can only adapt one answer so I adapted the earlier one... Your discrete example is wonderful!
                      $endgroup$
                      – Brent Wagner
                      2 hours ago
















                    4












                    $begingroup$

                    Here's a discrete example:
                    $$100P(x,y)=begin{array}{c|ccccc}xbackslash y&-2&-1&0&1&2\ hline -2&1&0&6&0&1\-1&0&9&0&9&0\0&6&0&36&0&6\1&0&9&0&9&0\2&1&0&6&0&1end{array}$$
                    Condition on any particular $x$, and $y$ has mean zero and variance $1$. Condition on any particular $y$, and $x$ has mean zero and variance $1$. But, of course, they're not independent, as the distribution's support isn't a Cartesian product.






                    share|cite|improve this answer









                    $endgroup$













                    • $begingroup$
                      Thank you very much! But I can only adapt one answer so I adapted the earlier one... Your discrete example is wonderful!
                      $endgroup$
                      – Brent Wagner
                      2 hours ago














                    4












                    4








                    4





                    $begingroup$

                    Here's a discrete example:
                    $$100P(x,y)=begin{array}{c|ccccc}xbackslash y&-2&-1&0&1&2\ hline -2&1&0&6&0&1\-1&0&9&0&9&0\0&6&0&36&0&6\1&0&9&0&9&0\2&1&0&6&0&1end{array}$$
                    Condition on any particular $x$, and $y$ has mean zero and variance $1$. Condition on any particular $y$, and $x$ has mean zero and variance $1$. But, of course, they're not independent, as the distribution's support isn't a Cartesian product.






                    share|cite|improve this answer









                    $endgroup$



                    Here's a discrete example:
                    $$100P(x,y)=begin{array}{c|ccccc}xbackslash y&-2&-1&0&1&2\ hline -2&1&0&6&0&1\-1&0&9&0&9&0\0&6&0&36&0&6\1&0&9&0&9&0\2&1&0&6&0&1end{array}$$
                    Condition on any particular $x$, and $y$ has mean zero and variance $1$. Condition on any particular $y$, and $x$ has mean zero and variance $1$. But, of course, they're not independent, as the distribution's support isn't a Cartesian product.







                    share|cite|improve this answer












                    share|cite|improve this answer



                    share|cite|improve this answer










                    answered 3 hours ago









                    jmerryjmerry

                    4,982514




                    4,982514












                    • $begingroup$
                      Thank you very much! But I can only adapt one answer so I adapted the earlier one... Your discrete example is wonderful!
                      $endgroup$
                      – Brent Wagner
                      2 hours ago


















                    • $begingroup$
                      Thank you very much! But I can only adapt one answer so I adapted the earlier one... Your discrete example is wonderful!
                      $endgroup$
                      – Brent Wagner
                      2 hours ago
















                    $begingroup$
                    Thank you very much! But I can only adapt one answer so I adapted the earlier one... Your discrete example is wonderful!
                    $endgroup$
                    – Brent Wagner
                    2 hours ago




                    $begingroup$
                    Thank you very much! But I can only adapt one answer so I adapted the earlier one... Your discrete example is wonderful!
                    $endgroup$
                    – Brent Wagner
                    2 hours ago










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










                    draft saved

                    draft discarded


















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













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












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
















                    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%2f3086758%2fif-the-expectation-and-variance-of-x-are-both-not-affected-by-y-and-vice-versa%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

                    Olav Thon

                    Image
                    Olav Thon Fødd 29. juni 1923 Ål Yrke forretningsdrivande Olav Thon på Commons Olav Thon (fødd 29. juni 1923) er ein norsk, stort sett sjølvlært forretningsmann, fødd i Ål i Hallingdal, som har gjort store pengar på særleg kjøp, bygging og utvikling av fast eigedom, mellom anna kjøpesentre, hotell og restaurantar, i både Noreg og somme andre land. Han vart Ridder av 1. klasse av St. Olavs Orden i 2003, og er som ivrig friluftsmann æresmedlem i Den Norske Turistforening etter å ha ytt store økonomiske bidrag til foreininga sine hytter og ruter i fjellet. Han er kjend for alltid å gå med ei karakteristisk strikkelue. Olav Thon i samtale med Trond Helleland på sentralbanksjefens årstale i februar 2016 I ung alder reiste Olav Thon til Newcastle for å lære seg engelsk, og han var seinare ei tid i Sveits, der han gjorde forretningar i import og eksport. Deretter tok han til å kjøpe faste eigedomar i Noreg med stor økonomisk suksess, verksemda voks snøgt og v...
                    Read more

                    Waikiki

                    Image
                    Denne artikkelen kan ha godt av ein språkvask , som reinskar opp målføringa og/eller innfører same språkstilen overalt. Denne artikkelen kan ha godt av ei opprydding for å nå ein høgare standard og/eller for å verta i tråd med standardoppsettet. Sjå korleis du redigerer ei side og stilmanualen for hjelp. Waikiki Land   USA Stat Hawaii Stad Honolulu Wikimedia Commons: Waikiki Utsyn over Waikiki Waikīkī er ein bydel i Honolulu, sør på øya Oahu, som er ei av Hawaii-øyane. Waikīkīstranda er ein av dei meste berømte strendene i verda. Waikīkī strekkjer seg langs stillehavskysten og har ei rekkje velrenommerte hotell. Bydelen er Hawaii-turismens høyborg. På Waikīkī Beach står ei statue av den lokale sportsstjerna Duka Kahanamoku. Kjelder | Denne artikkelen bygger på «Waikiki» frå Wikipedia på bokmål, den 4. november 2018. Denne artikkelen manglar kjelder eller referansar . Hjelp Wikipedia med å finna truverdige kjelder...
                    Read more

                    Tårekanal

                    Image
                    Tåreapparatet ( a ) Tårekjertel ( b ) Øvre tårepunkt ( c ) Øvre tårekanalikkel ( d ) Tåresekk ( e ) Nedre tårepunkt ( f ) Nedre tårekanalikkel ( g ) Tårekanal Tårekanalen eller nase-tårekanalen (av latin ductus nasolacrimalis ) er ein ductus som frakter overflødige tårer frå tåresekken til nasehola. Opninga inn mot nedre nasegang er delvis tildekt av ei ufullstendig slimhinnefold kalla Hasners ventil. Overflødige tårer strøymer gjennom tårekanalen og trekker inn i nedre nasegang. Dette er grunnen til at ein kan få rennande nase når ein gret, og at ein somme gonger kan kjenne saltsmaken av tårer i munnen. Både tårekanalen og tåresekken er tekt med fleirlaga sylinderepitel som inneheld slimproduserande begerceller, og er begge omgjevne av bindevev. Dakryostenose (tårekanalstenose) er ei innsnevring av tårekanalen som fører til epifora, ei overflod av tårer. Medfødd dakryostenose kan føra til cystisk utviding av tårekanalen ved at det dannast ein dakryocystocele. Personar m...
                    Read more