Three questions about the expectation of Brownian Motion. [on hold]











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Let W(r) is a standard brownian motion which follows N(0,r).



Then, how to calculate



(1) $E(int^1_0W(r)^3dr)$



(2) $E(int^1_0W(r)dr)^2$



(3) $E(W(1)int^1_0W(r)dr)^2$



For (1), my solution is $E(int^1_0W(r)^3dr)$ = $int^1_0E(W(r)^3)dr$ = $int^1_00dr$ = 0.



But, I cannot get idea about others.



Thanks for your time and consideration.






expected-value







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put on hold as too broad by amWhy, user10354138, Trevor Gunn, Cesareo, Chinnapparaj R 2 days ago


Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.



















    up vote
    0
    down vote

    favorite












    Let W(r) is a standard brownian motion which follows N(0,r).



    Then, how to calculate



    (1) $E(int^1_0W(r)^3dr)$



    (2) $E(int^1_0W(r)dr)^2$



    (3) $E(W(1)int^1_0W(r)dr)^2$



    For (1), my solution is $E(int^1_0W(r)^3dr)$ = $int^1_0E(W(r)^3)dr$ = $int^1_00dr$ = 0.



    But, I cannot get idea about others.



    Thanks for your time and consideration.






    expected-value







    share|cite|improve this question







    New contributor




    Kyoung Hoon Lee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.











    put on hold as too broad by amWhy, user10354138, Trevor Gunn, Cesareo, Chinnapparaj R 2 days ago


    Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

















      up vote
      0
      down vote

      favorite









      up vote
      0
      down vote

      favorite











      Let W(r) is a standard brownian motion which follows N(0,r).



      Then, how to calculate



      (1) $E(int^1_0W(r)^3dr)$



      (2) $E(int^1_0W(r)dr)^2$



      (3) $E(W(1)int^1_0W(r)dr)^2$



      For (1), my solution is $E(int^1_0W(r)^3dr)$ = $int^1_0E(W(r)^3)dr$ = $int^1_00dr$ = 0.



      But, I cannot get idea about others.



      Thanks for your time and consideration.






      expected-value







      share|cite|improve this question







      New contributor




      Kyoung Hoon Lee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      Let W(r) is a standard brownian motion which follows N(0,r).



      Then, how to calculate



      (1) $E(int^1_0W(r)^3dr)$



      (2) $E(int^1_0W(r)dr)^2$



      (3) $E(W(1)int^1_0W(r)dr)^2$



      For (1), my solution is $E(int^1_0W(r)^3dr)$ = $int^1_0E(W(r)^3)dr$ = $int^1_00dr$ = 0.



      But, I cannot get idea about others.



      Thanks for your time and consideration.






      expected-value




      expected-value






      share|cite|improve this question







      New contributor




      Kyoung Hoon Lee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question







      New contributor




      Kyoung Hoon Lee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




      share|cite|improve this question






      New contributor




      Kyoung Hoon Lee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      asked Nov 16 at 0:48









      Kyoung Hoon Lee

      1




      1




      New contributor




      Kyoung Hoon Lee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.





      New contributor





      Kyoung Hoon Lee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.






      Kyoung Hoon Lee is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.




      put on hold as too broad by amWhy, user10354138, Trevor Gunn, Cesareo, Chinnapparaj R 2 days ago


      Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.






      put on hold as too broad by amWhy, user10354138, Trevor Gunn, Cesareo, Chinnapparaj R 2 days ago


      Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.
























          1 Answer
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          Hints: $E(int_0^{1}W(r)dr)^{2}=Eint_0^{1}int_0^{1}W(r)W(s)dr ds=int_0^{1}int_0^{1}min{r,s} drds$ and a similar method gives 3): compute $EW(1)^{2}W(r)W(s)$ and the integrate.






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            1 Answer
            1






            active

            oldest

            votes








            1 Answer
            1






            active

            oldest

            votes









            active

            oldest

            votes






            active

            oldest

            votes








            up vote
            1
            down vote













            Hints: $E(int_0^{1}W(r)dr)^{2}=Eint_0^{1}int_0^{1}W(r)W(s)dr ds=int_0^{1}int_0^{1}min{r,s} drds$ and a similar method gives 3): compute $EW(1)^{2}W(r)W(s)$ and the integrate.






            share|cite|improve this answer

























              up vote
              1
              down vote













              Hints: $E(int_0^{1}W(r)dr)^{2}=Eint_0^{1}int_0^{1}W(r)W(s)dr ds=int_0^{1}int_0^{1}min{r,s} drds$ and a similar method gives 3): compute $EW(1)^{2}W(r)W(s)$ and the integrate.






              share|cite|improve this answer























                up vote
                1
                down vote










                up vote
                1
                down vote









                Hints: $E(int_0^{1}W(r)dr)^{2}=Eint_0^{1}int_0^{1}W(r)W(s)dr ds=int_0^{1}int_0^{1}min{r,s} drds$ and a similar method gives 3): compute $EW(1)^{2}W(r)W(s)$ and the integrate.






                share|cite|improve this answer












                Hints: $E(int_0^{1}W(r)dr)^{2}=Eint_0^{1}int_0^{1}W(r)W(s)dr ds=int_0^{1}int_0^{1}min{r,s} drds$ and a similar method gives 3): compute $EW(1)^{2}W(r)W(s)$ and the integrate.







                share|cite|improve this answer












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                share|cite|improve this answer










                answered 2 days ago









                Kavi Rama Murthy

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                39.7k31749















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