Prove $operatorname{Var}(X+Y) le 2operatorname{Var}(X) + 2operatorname{Var}(Y)$ where $X$ and $Y$ are not...











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I know how to show that $operatorname{Var}(X+Y) = operatorname{Var}(X) + operatorname{Var}(Y) + 2operatorname{Cov}(X,Y)$



I was also given the hint that I should use the triangle inequality to get



$|operatorname{Var}(X) + operatorname{Var}(Y) + 2operatorname{Cov}(X,Y)| le |operatorname{Var}(x)| + |operatorname{Var}(Y)| + |2operatorname{Cov}(X,Y)|$



Honestly, I have no idea where to go from here.










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    up vote
    1
    down vote

    favorite












    I know how to show that $operatorname{Var}(X+Y) = operatorname{Var}(X) + operatorname{Var}(Y) + 2operatorname{Cov}(X,Y)$



    I was also given the hint that I should use the triangle inequality to get



    $|operatorname{Var}(X) + operatorname{Var}(Y) + 2operatorname{Cov}(X,Y)| le |operatorname{Var}(x)| + |operatorname{Var}(Y)| + |2operatorname{Cov}(X,Y)|$



    Honestly, I have no idea where to go from here.










    share|cite|improve this question


























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I know how to show that $operatorname{Var}(X+Y) = operatorname{Var}(X) + operatorname{Var}(Y) + 2operatorname{Cov}(X,Y)$



      I was also given the hint that I should use the triangle inequality to get



      $|operatorname{Var}(X) + operatorname{Var}(Y) + 2operatorname{Cov}(X,Y)| le |operatorname{Var}(x)| + |operatorname{Var}(Y)| + |2operatorname{Cov}(X,Y)|$



      Honestly, I have no idea where to go from here.










      share|cite|improve this question















      I know how to show that $operatorname{Var}(X+Y) = operatorname{Var}(X) + operatorname{Var}(Y) + 2operatorname{Cov}(X,Y)$



      I was also given the hint that I should use the triangle inequality to get



      $|operatorname{Var}(X) + operatorname{Var}(Y) + 2operatorname{Cov}(X,Y)| le |operatorname{Var}(x)| + |operatorname{Var}(Y)| + |2operatorname{Cov}(X,Y)|$



      Honestly, I have no idea where to go from here.







      statistics covariance variance






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      edited Oct 11 at 21:38









      Bernard

      117k637110




      117k637110










      asked Oct 11 at 21:32









      mattfdz

      61




      61






















          5 Answers
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          The difference is
          $$2E((X-mu_X)^2)+2E((Y-mu_Y)^2)-E((X+Y-mu_X-mu_Y)^2)
          =2E(X'^2)+2E(Y'^2)-E((X'+Y')^2)$$

          where $X'=X-mu_X$, $Y'=Y-mu_Y$. Can you simplify that to something conveniently positive?






          share|cite|improve this answer




























            up vote
            0
            down vote













            The key is to show that $|2 operatorname{Cov}(X,Y)| leq operatorname{Var}(X) + operatorname{Var}(Y)$.



            Here's a hint towards that: Write $hat{X} = X - mu_X$ and $hat{Y} = Y - mu_Y$. Then note that $$E[(hat{X} - hat{Y})^2] geq 0,.$$



            What happens if you expand the expectation?






            share|cite|improve this answer






























              up vote
              0
              down vote













              Let $X'=X-EX,Y'=Y-EY$. Then the inequality to be proved is $E(X'+Y')^{2} leq 2EX'^{2}+2EY'^{2}$ whic follows from the fact that $(a+b)^{2} leq 2a^{2} +2b^{2}$ for any two real numbers $a$ and $b$.






              share|cite|improve this answer




























                up vote
                0
                down vote













                Start as you did with:
                $$
                Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y).
                $$

                The Cauchy–Schwarz inequality then gives us:
                $$
                2Cov(X, Y) leq 2[Var(X)]^{1/2}[Var(Y)]^{1/2}.
                $$

                Finally, Young's inequality for products gives us:
                $$
                2[Var(X)]^{1/2}[Var(Y)]^{1/2} leq Var(X) + Var(Y),
                $$

                so we get, as desired:
                $$
                Var(X + Y) leq 2Var(X) + 2Var(Y).
                $$






                share|cite|improve this answer




























                  up vote
                  0
                  down vote













                  If you know how to show
                  $$operatorname{Var}(X+Y) = operatorname{Var}(X) + operatorname{Var}(Y) + 2operatorname{Cov}(X,Y)tag1$$
                  then you can show
                  $$
                  operatorname{Var}(X-Y) = operatorname{Var}(X) + operatorname{Var}(Y) - 2operatorname{Cov}(X,Y).tag2
                  $$

                  Now add equations (1) and (2) to get
                  $$
                  operatorname{Var}(X+Y)+operatorname{Var}(X-Y) = 2operatorname{Var}(X) + 2operatorname{Var}(Y).
                  $$

                  Finally the fact that variance of anything is non-negative implies that
                  $$
                  operatorname{Var}(X+Y)le operatorname{Var}(X+Y)+operatorname{Var}(X-Y).
                  $$






                  share|cite|improve this answer





















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                    5 Answers
                    5






                    active

                    oldest

                    votes








                    5 Answers
                    5






                    active

                    oldest

                    votes









                    active

                    oldest

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                    active

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                    up vote
                    0
                    down vote













                    The difference is
                    $$2E((X-mu_X)^2)+2E((Y-mu_Y)^2)-E((X+Y-mu_X-mu_Y)^2)
                    =2E(X'^2)+2E(Y'^2)-E((X'+Y')^2)$$

                    where $X'=X-mu_X$, $Y'=Y-mu_Y$. Can you simplify that to something conveniently positive?






                    share|cite|improve this answer

























                      up vote
                      0
                      down vote













                      The difference is
                      $$2E((X-mu_X)^2)+2E((Y-mu_Y)^2)-E((X+Y-mu_X-mu_Y)^2)
                      =2E(X'^2)+2E(Y'^2)-E((X'+Y')^2)$$

                      where $X'=X-mu_X$, $Y'=Y-mu_Y$. Can you simplify that to something conveniently positive?






                      share|cite|improve this answer























                        up vote
                        0
                        down vote










                        up vote
                        0
                        down vote









                        The difference is
                        $$2E((X-mu_X)^2)+2E((Y-mu_Y)^2)-E((X+Y-mu_X-mu_Y)^2)
                        =2E(X'^2)+2E(Y'^2)-E((X'+Y')^2)$$

                        where $X'=X-mu_X$, $Y'=Y-mu_Y$. Can you simplify that to something conveniently positive?






                        share|cite|improve this answer












                        The difference is
                        $$2E((X-mu_X)^2)+2E((Y-mu_Y)^2)-E((X+Y-mu_X-mu_Y)^2)
                        =2E(X'^2)+2E(Y'^2)-E((X'+Y')^2)$$

                        where $X'=X-mu_X$, $Y'=Y-mu_Y$. Can you simplify that to something conveniently positive?







                        share|cite|improve this answer












                        share|cite|improve this answer



                        share|cite|improve this answer










                        answered Oct 11 at 21:36









                        Lord Shark the Unknown

                        100k958131




                        100k958131






















                            up vote
                            0
                            down vote













                            The key is to show that $|2 operatorname{Cov}(X,Y)| leq operatorname{Var}(X) + operatorname{Var}(Y)$.



                            Here's a hint towards that: Write $hat{X} = X - mu_X$ and $hat{Y} = Y - mu_Y$. Then note that $$E[(hat{X} - hat{Y})^2] geq 0,.$$



                            What happens if you expand the expectation?






                            share|cite|improve this answer



























                              up vote
                              0
                              down vote













                              The key is to show that $|2 operatorname{Cov}(X,Y)| leq operatorname{Var}(X) + operatorname{Var}(Y)$.



                              Here's a hint towards that: Write $hat{X} = X - mu_X$ and $hat{Y} = Y - mu_Y$. Then note that $$E[(hat{X} - hat{Y})^2] geq 0,.$$



                              What happens if you expand the expectation?






                              share|cite|improve this answer

























                                up vote
                                0
                                down vote










                                up vote
                                0
                                down vote









                                The key is to show that $|2 operatorname{Cov}(X,Y)| leq operatorname{Var}(X) + operatorname{Var}(Y)$.



                                Here's a hint towards that: Write $hat{X} = X - mu_X$ and $hat{Y} = Y - mu_Y$. Then note that $$E[(hat{X} - hat{Y})^2] geq 0,.$$



                                What happens if you expand the expectation?






                                share|cite|improve this answer














                                The key is to show that $|2 operatorname{Cov}(X,Y)| leq operatorname{Var}(X) + operatorname{Var}(Y)$.



                                Here's a hint towards that: Write $hat{X} = X - mu_X$ and $hat{Y} = Y - mu_Y$. Then note that $$E[(hat{X} - hat{Y})^2] geq 0,.$$



                                What happens if you expand the expectation?







                                share|cite|improve this answer














                                share|cite|improve this answer



                                share|cite|improve this answer








                                edited Oct 11 at 21:39









                                Bernard

                                117k637110




                                117k637110










                                answered Oct 11 at 21:38









                                Marcus M

                                8,7281947




                                8,7281947






















                                    up vote
                                    0
                                    down vote













                                    Let $X'=X-EX,Y'=Y-EY$. Then the inequality to be proved is $E(X'+Y')^{2} leq 2EX'^{2}+2EY'^{2}$ whic follows from the fact that $(a+b)^{2} leq 2a^{2} +2b^{2}$ for any two real numbers $a$ and $b$.






                                    share|cite|improve this answer

























                                      up vote
                                      0
                                      down vote













                                      Let $X'=X-EX,Y'=Y-EY$. Then the inequality to be proved is $E(X'+Y')^{2} leq 2EX'^{2}+2EY'^{2}$ whic follows from the fact that $(a+b)^{2} leq 2a^{2} +2b^{2}$ for any two real numbers $a$ and $b$.






                                      share|cite|improve this answer























                                        up vote
                                        0
                                        down vote










                                        up vote
                                        0
                                        down vote









                                        Let $X'=X-EX,Y'=Y-EY$. Then the inequality to be proved is $E(X'+Y')^{2} leq 2EX'^{2}+2EY'^{2}$ whic follows from the fact that $(a+b)^{2} leq 2a^{2} +2b^{2}$ for any two real numbers $a$ and $b$.






                                        share|cite|improve this answer












                                        Let $X'=X-EX,Y'=Y-EY$. Then the inequality to be proved is $E(X'+Y')^{2} leq 2EX'^{2}+2EY'^{2}$ whic follows from the fact that $(a+b)^{2} leq 2a^{2} +2b^{2}$ for any two real numbers $a$ and $b$.







                                        share|cite|improve this answer












                                        share|cite|improve this answer



                                        share|cite|improve this answer










                                        answered Oct 11 at 23:45









                                        Kavi Rama Murthy

                                        48.2k31854




                                        48.2k31854






















                                            up vote
                                            0
                                            down vote













                                            Start as you did with:
                                            $$
                                            Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y).
                                            $$

                                            The Cauchy–Schwarz inequality then gives us:
                                            $$
                                            2Cov(X, Y) leq 2[Var(X)]^{1/2}[Var(Y)]^{1/2}.
                                            $$

                                            Finally, Young's inequality for products gives us:
                                            $$
                                            2[Var(X)]^{1/2}[Var(Y)]^{1/2} leq Var(X) + Var(Y),
                                            $$

                                            so we get, as desired:
                                            $$
                                            Var(X + Y) leq 2Var(X) + 2Var(Y).
                                            $$






                                            share|cite|improve this answer

























                                              up vote
                                              0
                                              down vote













                                              Start as you did with:
                                              $$
                                              Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y).
                                              $$

                                              The Cauchy–Schwarz inequality then gives us:
                                              $$
                                              2Cov(X, Y) leq 2[Var(X)]^{1/2}[Var(Y)]^{1/2}.
                                              $$

                                              Finally, Young's inequality for products gives us:
                                              $$
                                              2[Var(X)]^{1/2}[Var(Y)]^{1/2} leq Var(X) + Var(Y),
                                              $$

                                              so we get, as desired:
                                              $$
                                              Var(X + Y) leq 2Var(X) + 2Var(Y).
                                              $$






                                              share|cite|improve this answer























                                                up vote
                                                0
                                                down vote










                                                up vote
                                                0
                                                down vote









                                                Start as you did with:
                                                $$
                                                Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y).
                                                $$

                                                The Cauchy–Schwarz inequality then gives us:
                                                $$
                                                2Cov(X, Y) leq 2[Var(X)]^{1/2}[Var(Y)]^{1/2}.
                                                $$

                                                Finally, Young's inequality for products gives us:
                                                $$
                                                2[Var(X)]^{1/2}[Var(Y)]^{1/2} leq Var(X) + Var(Y),
                                                $$

                                                so we get, as desired:
                                                $$
                                                Var(X + Y) leq 2Var(X) + 2Var(Y).
                                                $$






                                                share|cite|improve this answer












                                                Start as you did with:
                                                $$
                                                Var(X + Y) = Var(X) + Var(Y) + 2Cov(X, Y).
                                                $$

                                                The Cauchy–Schwarz inequality then gives us:
                                                $$
                                                2Cov(X, Y) leq 2[Var(X)]^{1/2}[Var(Y)]^{1/2}.
                                                $$

                                                Finally, Young's inequality for products gives us:
                                                $$
                                                2[Var(X)]^{1/2}[Var(Y)]^{1/2} leq Var(X) + Var(Y),
                                                $$

                                                so we get, as desired:
                                                $$
                                                Var(X + Y) leq 2Var(X) + 2Var(Y).
                                                $$







                                                share|cite|improve this answer












                                                share|cite|improve this answer



                                                share|cite|improve this answer










                                                answered Nov 29 at 5:22









                                                Fredrik Savje

                                                1304




                                                1304






















                                                    up vote
                                                    0
                                                    down vote













                                                    If you know how to show
                                                    $$operatorname{Var}(X+Y) = operatorname{Var}(X) + operatorname{Var}(Y) + 2operatorname{Cov}(X,Y)tag1$$
                                                    then you can show
                                                    $$
                                                    operatorname{Var}(X-Y) = operatorname{Var}(X) + operatorname{Var}(Y) - 2operatorname{Cov}(X,Y).tag2
                                                    $$

                                                    Now add equations (1) and (2) to get
                                                    $$
                                                    operatorname{Var}(X+Y)+operatorname{Var}(X-Y) = 2operatorname{Var}(X) + 2operatorname{Var}(Y).
                                                    $$

                                                    Finally the fact that variance of anything is non-negative implies that
                                                    $$
                                                    operatorname{Var}(X+Y)le operatorname{Var}(X+Y)+operatorname{Var}(X-Y).
                                                    $$






                                                    share|cite|improve this answer

























                                                      up vote
                                                      0
                                                      down vote













                                                      If you know how to show
                                                      $$operatorname{Var}(X+Y) = operatorname{Var}(X) + operatorname{Var}(Y) + 2operatorname{Cov}(X,Y)tag1$$
                                                      then you can show
                                                      $$
                                                      operatorname{Var}(X-Y) = operatorname{Var}(X) + operatorname{Var}(Y) - 2operatorname{Cov}(X,Y).tag2
                                                      $$

                                                      Now add equations (1) and (2) to get
                                                      $$
                                                      operatorname{Var}(X+Y)+operatorname{Var}(X-Y) = 2operatorname{Var}(X) + 2operatorname{Var}(Y).
                                                      $$

                                                      Finally the fact that variance of anything is non-negative implies that
                                                      $$
                                                      operatorname{Var}(X+Y)le operatorname{Var}(X+Y)+operatorname{Var}(X-Y).
                                                      $$






                                                      share|cite|improve this answer























                                                        up vote
                                                        0
                                                        down vote










                                                        up vote
                                                        0
                                                        down vote









                                                        If you know how to show
                                                        $$operatorname{Var}(X+Y) = operatorname{Var}(X) + operatorname{Var}(Y) + 2operatorname{Cov}(X,Y)tag1$$
                                                        then you can show
                                                        $$
                                                        operatorname{Var}(X-Y) = operatorname{Var}(X) + operatorname{Var}(Y) - 2operatorname{Cov}(X,Y).tag2
                                                        $$

                                                        Now add equations (1) and (2) to get
                                                        $$
                                                        operatorname{Var}(X+Y)+operatorname{Var}(X-Y) = 2operatorname{Var}(X) + 2operatorname{Var}(Y).
                                                        $$

                                                        Finally the fact that variance of anything is non-negative implies that
                                                        $$
                                                        operatorname{Var}(X+Y)le operatorname{Var}(X+Y)+operatorname{Var}(X-Y).
                                                        $$






                                                        share|cite|improve this answer












                                                        If you know how to show
                                                        $$operatorname{Var}(X+Y) = operatorname{Var}(X) + operatorname{Var}(Y) + 2operatorname{Cov}(X,Y)tag1$$
                                                        then you can show
                                                        $$
                                                        operatorname{Var}(X-Y) = operatorname{Var}(X) + operatorname{Var}(Y) - 2operatorname{Cov}(X,Y).tag2
                                                        $$

                                                        Now add equations (1) and (2) to get
                                                        $$
                                                        operatorname{Var}(X+Y)+operatorname{Var}(X-Y) = 2operatorname{Var}(X) + 2operatorname{Var}(Y).
                                                        $$

                                                        Finally the fact that variance of anything is non-negative implies that
                                                        $$
                                                        operatorname{Var}(X+Y)le operatorname{Var}(X+Y)+operatorname{Var}(X-Y).
                                                        $$







                                                        share|cite|improve this answer












                                                        share|cite|improve this answer



                                                        share|cite|improve this answer










                                                        answered Nov 29 at 14:56









                                                        grand_chat

                                                        19.9k11225




                                                        19.9k11225






























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