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












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










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    $begingroup$


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










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      $begingroup$


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










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







      probability






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      asked Dec 5 '18 at 20:05









      Will.ZWill.Z

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          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
          $$
          operatorname E|X-mu|leoperatorname E|X|+|mu|<infty.
          $$






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            $begingroup$

            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






























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