Finite expectation of a random variable [duplicate]












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  • Explain why $E(X) = int_0^infty (1-F_X (t)) , dt$ for every nonnegative random variable $X$

    3 answers




$X geq 0$ be a random variable defined on $(Omega,mathcal{F},P)$. Show that $mathbb{E}[X]<infty iff Sigma_{n=1}^infty P(X>n) < infty $.



I got the reverse direction but I am struggling with the $"implies"$ direction. So far, I have the following worked out:



$mathbb{E}[X]<infty$



$implies int_0^infty (1-F(x)) dx < infty$ (where $F$ is the distribution function of the random variable X)



$implies int_0^infty (1-P(Xleq x)) dx < infty$



$implies int_0^infty P(X>x) dx < infty$



Consider $int_0^infty P(X>x) dx$



$= Sigma_{n=1}^infty int_{n-1}^n P(X>x) dx$



This is the point I am stuck at. Any help will be deeply appreciated!






probability-theory random-variables expected-value







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 Dec 3 '18 at 19:14


This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.




















    0












    $begingroup$



    This question already has an answer here:




    • Explain why $E(X) = int_0^infty (1-F_X (t)) , dt$ for every nonnegative random variable $X$

      3 answers




    $X geq 0$ be a random variable defined on $(Omega,mathcal{F},P)$. Show that $mathbb{E}[X]<infty iff Sigma_{n=1}^infty P(X>n) < infty $.



    I got the reverse direction but I am struggling with the $"implies"$ direction. So far, I have the following worked out:



    $mathbb{E}[X]<infty$



    $implies int_0^infty (1-F(x)) dx < infty$ (where $F$ is the distribution function of the random variable X)



    $implies int_0^infty (1-P(Xleq x)) dx < infty$



    $implies int_0^infty P(X>x) dx < infty$



    Consider $int_0^infty P(X>x) dx$



    $= Sigma_{n=1}^infty int_{n-1}^n P(X>x) dx$



    This is the point I am stuck at. Any help will be deeply appreciated!






    probability-theory random-variables expected-value







    share|cite|improve this question









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     Dec 3 '18 at 19:14


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      $begingroup$



      This question already has an answer here:




      • Explain why $E(X) = int_0^infty (1-F_X (t)) , dt$ for every nonnegative random variable $X$

        3 answers




      $X geq 0$ be a random variable defined on $(Omega,mathcal{F},P)$. Show that $mathbb{E}[X]<infty iff Sigma_{n=1}^infty P(X>n) < infty $.



      I got the reverse direction but I am struggling with the $"implies"$ direction. So far, I have the following worked out:



      $mathbb{E}[X]<infty$



      $implies int_0^infty (1-F(x)) dx < infty$ (where $F$ is the distribution function of the random variable X)



      $implies int_0^infty (1-P(Xleq x)) dx < infty$



      $implies int_0^infty P(X>x) dx < infty$



      Consider $int_0^infty P(X>x) dx$



      $= Sigma_{n=1}^infty int_{n-1}^n P(X>x) dx$



      This is the point I am stuck at. Any help will be deeply appreciated!






      probability-theory random-variables expected-value







      share|cite|improve this question









      $endgroup$





      This question already has an answer here:




      • Explain why $E(X) = int_0^infty (1-F_X (t)) , dt$ for every nonnegative random variable $X$

        3 answers




      $X geq 0$ be a random variable defined on $(Omega,mathcal{F},P)$. Show that $mathbb{E}[X]<infty iff Sigma_{n=1}^infty P(X>n) < infty $.



      I got the reverse direction but I am struggling with the $"implies"$ direction. So far, I have the following worked out:



      $mathbb{E}[X]<infty$



      $implies int_0^infty (1-F(x)) dx < infty$ (where $F$ is the distribution function of the random variable X)



      $implies int_0^infty (1-P(Xleq x)) dx < infty$



      $implies int_0^infty P(X>x) dx < infty$



      Consider $int_0^infty P(X>x) dx$



      $= Sigma_{n=1}^infty int_{n-1}^n P(X>x) dx$



      This is the point I am stuck at. Any help will be deeply appreciated!





      This question already has an answer here:




      • Explain why $E(X) = int_0^infty (1-F_X (t)) , dt$ for every nonnegative random variable $X$

        3 answers







      probability-theory random-variables expected-value




      probability-theory random-variables expected-value






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 3 '18 at 16:44









      B.TB.T

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      marked as duplicate by Did probability-theory
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       Dec 3 '18 at 19:14


      This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.






      marked as duplicate by Did probability-theory
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          1 Answer
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          $begingroup$

          Hint: $x ge sum_{n=1}^infty I_{{x ge n}} ge x-1$ for $x ge 0$, where $I_A$ is the indicator of $A$ ($1$ when $A$ is true, $0$ when $A$ is false).






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            1 Answer
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            active

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            1 Answer
            1






            active

            oldest

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            active

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            active

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            0












            $begingroup$

            Hint: $x ge sum_{n=1}^infty I_{{x ge n}} ge x-1$ for $x ge 0$, where $I_A$ is the indicator of $A$ ($1$ when $A$ is true, $0$ when $A$ is false).






            share|cite|improve this answer









            $endgroup$


















              0












              $begingroup$

              Hint: $x ge sum_{n=1}^infty I_{{x ge n}} ge x-1$ for $x ge 0$, where $I_A$ is the indicator of $A$ ($1$ when $A$ is true, $0$ when $A$ is false).






              share|cite|improve this answer









              $endgroup$
















                0












                0








                0





                $begingroup$

                Hint: $x ge sum_{n=1}^infty I_{{x ge n}} ge x-1$ for $x ge 0$, where $I_A$ is the indicator of $A$ ($1$ when $A$ is true, $0$ when $A$ is false).






                share|cite|improve this answer









                $endgroup$



                Hint: $x ge sum_{n=1}^infty I_{{x ge n}} ge x-1$ for $x ge 0$, where $I_A$ is the indicator of $A$ ($1$ when $A$ is true, $0$ when $A$ is false).







                share|cite|improve this answer












                share|cite|improve this answer



                share|cite|improve this answer










                answered Dec 3 '18 at 16:49









                Robert IsraelRobert Israel

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                320k23209459















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