Expectation of absolute value of the difference between a random variable and its mean












1












$begingroup$


Suppose $Y$ is a random variable with finite mean $mu$. Then is $mathbb{E}(|Y-mu|)$ finite or not? Why?










share|cite|improve this question









$endgroup$

















    1












    $begingroup$


    Suppose $Y$ is a random variable with finite mean $mu$. Then is $mathbb{E}(|Y-mu|)$ finite or not? Why?










    share|cite|improve this question









    $endgroup$















      1












      1








      1





      $begingroup$


      Suppose $Y$ is a random variable with finite mean $mu$. Then is $mathbb{E}(|Y-mu|)$ finite or not? Why?










      share|cite|improve this question









      $endgroup$




      Suppose $Y$ is a random variable with finite mean $mu$. Then is $mathbb{E}(|Y-mu|)$ finite or not? Why?







      probability






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 5 '18 at 20:05









      Will.ZWill.Z

      133




      133






















          1 Answer
          1






          active

          oldest

          votes


















          0












          $begingroup$

          The expected value of a random variable $X$ exists and is finite if both $operatorname EX^+$ and $operatorname EX^{-}$ are finite, where $X^+(omega)=max{X(omega),0}$ and $X^-(omega)=-min{X(omega),0}$ for $omegainOmega$. We have that $operatorname EX=operatorname EX^+-operatorname EX^{-}$. If the expected value is finite, the first absolute moment is also finite since $E|X|=operatorname EX^++operatorname EX^-$. It follows that
          $$
          operatorname E|X-mu|leoperatorname E|X|+|mu|<infty.
          $$






          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%2f3027577%2fexpectation-of-absolute-value-of-the-difference-between-a-random-variable-and-it%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









            0












            $begingroup$

            The expected value of a random variable $X$ exists and is finite if both $operatorname EX^+$ and $operatorname EX^{-}$ are finite, where $X^+(omega)=max{X(omega),0}$ and $X^-(omega)=-min{X(omega),0}$ for $omegainOmega$. We have that $operatorname EX=operatorname EX^+-operatorname EX^{-}$. If the expected value is finite, the first absolute moment is also finite since $E|X|=operatorname EX^++operatorname EX^-$. It follows that
            $$
            operatorname E|X-mu|leoperatorname E|X|+|mu|<infty.
            $$






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              The expected value of a random variable $X$ exists and is finite if both $operatorname EX^+$ and $operatorname EX^{-}$ are finite, where $X^+(omega)=max{X(omega),0}$ and $X^-(omega)=-min{X(omega),0}$ for $omegainOmega$. We have that $operatorname EX=operatorname EX^+-operatorname EX^{-}$. If the expected value is finite, the first absolute moment is also finite since $E|X|=operatorname EX^++operatorname EX^-$. It follows that
              $$
              operatorname E|X-mu|leoperatorname E|X|+|mu|<infty.
              $$






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                The expected value of a random variable $X$ exists and is finite if both $operatorname EX^+$ and $operatorname EX^{-}$ are finite, where $X^+(omega)=max{X(omega),0}$ and $X^-(omega)=-min{X(omega),0}$ for $omegainOmega$. We have that $operatorname EX=operatorname EX^+-operatorname EX^{-}$. If the expected value is finite, the first absolute moment is also finite since $E|X|=operatorname EX^++operatorname EX^-$. It follows that
                $$
                operatorname E|X-mu|leoperatorname E|X|+|mu|<infty.
                $$






                share|cite|improve this answer









                $endgroup$



                The expected value of a random variable $X$ exists and is finite if both $operatorname EX^+$ and $operatorname EX^{-}$ are finite, where $X^+(omega)=max{X(omega),0}$ and $X^-(omega)=-min{X(omega),0}$ for $omegainOmega$. We have that $operatorname EX=operatorname EX^+-operatorname EX^{-}$. If the expected value is finite, the first absolute moment is also finite since $E|X|=operatorname EX^++operatorname EX^-$. It follows that
                $$
                operatorname E|X-mu|leoperatorname E|X|+|mu|<infty.
                $$







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 5 '18 at 20:11









                Cm7F7BbCm7F7Bb

                12.5k32243




                12.5k32243






























                    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%2f3027577%2fexpectation-of-absolute-value-of-the-difference-between-a-random-variable-and-it%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

                    How to change which sound is reproduced for terminal bell?

                    Title Spacing in Bjornstrup Chapter, Removing Chapter Number From Contents

                    Can I use Tabulator js library in my java Spring + Thymeleaf project?