Show that the expected value of $X$ is equal to $E(x)$











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Consider a random variable $X : Ω → {1, . . . , 15}$, that takes on integer values in the set ${1, . . . , 15}$. Show that the expected value of $X$ is equal to: $$E(X) = sum_{i=1}^{15} P(X ≥ i)$$











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    Consider a random variable $X : Ω → {1, . . . , 15}$, that takes on integer values in the set ${1, . . . , 15}$. Show that the expected value of $X$ is equal to: $$E(X) = sum_{i=1}^{15} P(X ≥ i)$$











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      up vote
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      up vote
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      down vote

      favorite












      Consider a random variable $X : Ω → {1, . . . , 15}$, that takes on integer values in the set ${1, . . . , 15}$. Show that the expected value of $X$ is equal to: $$E(X) = sum_{i=1}^{15} P(X ≥ i)$$











      share|cite|improve this question
















      Consider a random variable $X : Ω → {1, . . . , 15}$, that takes on integer values in the set ${1, . . . , 15}$. Show that the expected value of $X$ is equal to: $$E(X) = sum_{i=1}^{15} P(X ≥ i)$$








      probability summation random-variables expected-value






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      edited Nov 22 at 16:47









      greedoid

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      asked Nov 22 at 12:45









      Man

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          begin{eqnarray}
          E(X)&=& sum_{k=1}^{15}k cdot mathbb P{X=k}\
          &=& color{red}{mathbb P{X=1}}\
          &+& color{red}{mathbb P{X=2}}+ color{blue}{mathbb P{X=2}}\
          &+& color{red}{mathbb P{X=3}}+ color{blue}{mathbb P{X=3}}+ color{green}{mathbb P{X=3}}\
          & vdots &\
          &+& color{red}{mathbb P{X=15}}+ color{blue}{mathbb P{X=15}}+ color{green}{mathbb P{X=15}}+dots + mathbb P{X=15}\
          &=& color{red}{mathbb P{Xgeq 1}}+ color{blue}{mathbb P{Xgeq 2}}+ color{green}{mathbb P{Xgeq 3}}+dots + mathbb P{X=15}\
          &=& sum_{k=1}^{15} mathbb P{Xgeq k}\
          end{eqnarray}






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            Hint



            $$mathbb P{Xgeq i}=sum_{k=i}^{15}mathbb P{X=k}.$$






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              2 Answers
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              begin{eqnarray}
              E(X)&=& sum_{k=1}^{15}k cdot mathbb P{X=k}\
              &=& color{red}{mathbb P{X=1}}\
              &+& color{red}{mathbb P{X=2}}+ color{blue}{mathbb P{X=2}}\
              &+& color{red}{mathbb P{X=3}}+ color{blue}{mathbb P{X=3}}+ color{green}{mathbb P{X=3}}\
              & vdots &\
              &+& color{red}{mathbb P{X=15}}+ color{blue}{mathbb P{X=15}}+ color{green}{mathbb P{X=15}}+dots + mathbb P{X=15}\
              &=& color{red}{mathbb P{Xgeq 1}}+ color{blue}{mathbb P{Xgeq 2}}+ color{green}{mathbb P{Xgeq 3}}+dots + mathbb P{X=15}\
              &=& sum_{k=1}^{15} mathbb P{Xgeq k}\
              end{eqnarray}






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                begin{eqnarray}
                E(X)&=& sum_{k=1}^{15}k cdot mathbb P{X=k}\
                &=& color{red}{mathbb P{X=1}}\
                &+& color{red}{mathbb P{X=2}}+ color{blue}{mathbb P{X=2}}\
                &+& color{red}{mathbb P{X=3}}+ color{blue}{mathbb P{X=3}}+ color{green}{mathbb P{X=3}}\
                & vdots &\
                &+& color{red}{mathbb P{X=15}}+ color{blue}{mathbb P{X=15}}+ color{green}{mathbb P{X=15}}+dots + mathbb P{X=15}\
                &=& color{red}{mathbb P{Xgeq 1}}+ color{blue}{mathbb P{Xgeq 2}}+ color{green}{mathbb P{Xgeq 3}}+dots + mathbb P{X=15}\
                &=& sum_{k=1}^{15} mathbb P{Xgeq k}\
                end{eqnarray}






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                  up vote
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                  up vote
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                  begin{eqnarray}
                  E(X)&=& sum_{k=1}^{15}k cdot mathbb P{X=k}\
                  &=& color{red}{mathbb P{X=1}}\
                  &+& color{red}{mathbb P{X=2}}+ color{blue}{mathbb P{X=2}}\
                  &+& color{red}{mathbb P{X=3}}+ color{blue}{mathbb P{X=3}}+ color{green}{mathbb P{X=3}}\
                  & vdots &\
                  &+& color{red}{mathbb P{X=15}}+ color{blue}{mathbb P{X=15}}+ color{green}{mathbb P{X=15}}+dots + mathbb P{X=15}\
                  &=& color{red}{mathbb P{Xgeq 1}}+ color{blue}{mathbb P{Xgeq 2}}+ color{green}{mathbb P{Xgeq 3}}+dots + mathbb P{X=15}\
                  &=& sum_{k=1}^{15} mathbb P{Xgeq k}\
                  end{eqnarray}






                  share|cite|improve this answer












                  begin{eqnarray}
                  E(X)&=& sum_{k=1}^{15}k cdot mathbb P{X=k}\
                  &=& color{red}{mathbb P{X=1}}\
                  &+& color{red}{mathbb P{X=2}}+ color{blue}{mathbb P{X=2}}\
                  &+& color{red}{mathbb P{X=3}}+ color{blue}{mathbb P{X=3}}+ color{green}{mathbb P{X=3}}\
                  & vdots &\
                  &+& color{red}{mathbb P{X=15}}+ color{blue}{mathbb P{X=15}}+ color{green}{mathbb P{X=15}}+dots + mathbb P{X=15}\
                  &=& color{red}{mathbb P{Xgeq 1}}+ color{blue}{mathbb P{Xgeq 2}}+ color{green}{mathbb P{Xgeq 3}}+dots + mathbb P{X=15}\
                  &=& sum_{k=1}^{15} mathbb P{Xgeq k}\
                  end{eqnarray}







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                  answered Nov 22 at 13:07









                  greedoid

                  36.5k114592




                  36.5k114592






















                      up vote
                      0
                      down vote













                      Hint



                      $$mathbb P{Xgeq i}=sum_{k=i}^{15}mathbb P{X=k}.$$






                      share|cite|improve this answer

























                        up vote
                        0
                        down vote













                        Hint



                        $$mathbb P{Xgeq i}=sum_{k=i}^{15}mathbb P{X=k}.$$






                        share|cite|improve this answer























                          up vote
                          0
                          down vote










                          up vote
                          0
                          down vote









                          Hint



                          $$mathbb P{Xgeq i}=sum_{k=i}^{15}mathbb P{X=k}.$$






                          share|cite|improve this answer












                          Hint



                          $$mathbb P{Xgeq i}=sum_{k=i}^{15}mathbb P{X=k}.$$







                          share|cite|improve this answer












                          share|cite|improve this answer



                          share|cite|improve this answer










                          answered Nov 22 at 12:49









                          Surb

                          37.3k94375




                          37.3k94375






























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