Finding $P(X_1+X_2+X_3+X_4ge3)$ for independent $X_isim U(0,1)$












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How to find $P(X_1 + X_2 + X_3 + X_4 geq 3)$ for uniformly distributed independent random variables $X_1$, $X_2$, $X_3$, $X_4sim U(0,1)$?



It follows from independence that their cumulative density function is 1, but I'm struggling with integration space.










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  • 1




    $begingroup$
    (Big) Hint: By symmetry, this is equal to $$P(X_1+X_2+X_3+X_4le1)$$ Now, compute the latter...
    $endgroup$
    – Did
    Dec 4 '18 at 15:23










  • $begingroup$
    @Did I can't figure that out. Looks like it gets us to equation $$F(1) = 1 - F(3)$$ but I can't recall any formal symmetry connected with that,
    $endgroup$
    – AromaTheLoop
    Dec 4 '18 at 17:07






  • 1




    $begingroup$
    Sub-hint: $(1-X_1,1-X_2,1-X_3,1-X_4)$ is distributed like $(X_1,X_2,X_3,X_4)$.
    $endgroup$
    – Did
    Dec 4 '18 at 17:36










  • $begingroup$
    @Did Oh I see, thanks!
    $endgroup$
    – AromaTheLoop
    Dec 4 '18 at 17:54


















1












$begingroup$


How to find $P(X_1 + X_2 + X_3 + X_4 geq 3)$ for uniformly distributed independent random variables $X_1$, $X_2$, $X_3$, $X_4sim U(0,1)$?



It follows from independence that their cumulative density function is 1, but I'm struggling with integration space.










share|cite|improve this question











$endgroup$








  • 1




    $begingroup$
    (Big) Hint: By symmetry, this is equal to $$P(X_1+X_2+X_3+X_4le1)$$ Now, compute the latter...
    $endgroup$
    – Did
    Dec 4 '18 at 15:23










  • $begingroup$
    @Did I can't figure that out. Looks like it gets us to equation $$F(1) = 1 - F(3)$$ but I can't recall any formal symmetry connected with that,
    $endgroup$
    – AromaTheLoop
    Dec 4 '18 at 17:07






  • 1




    $begingroup$
    Sub-hint: $(1-X_1,1-X_2,1-X_3,1-X_4)$ is distributed like $(X_1,X_2,X_3,X_4)$.
    $endgroup$
    – Did
    Dec 4 '18 at 17:36










  • $begingroup$
    @Did Oh I see, thanks!
    $endgroup$
    – AromaTheLoop
    Dec 4 '18 at 17:54
















1












1








1


0



$begingroup$


How to find $P(X_1 + X_2 + X_3 + X_4 geq 3)$ for uniformly distributed independent random variables $X_1$, $X_2$, $X_3$, $X_4sim U(0,1)$?



It follows from independence that their cumulative density function is 1, but I'm struggling with integration space.










share|cite|improve this question











$endgroup$




How to find $P(X_1 + X_2 + X_3 + X_4 geq 3)$ for uniformly distributed independent random variables $X_1$, $X_2$, $X_3$, $X_4sim U(0,1)$?



It follows from independence that their cumulative density function is 1, but I'm struggling with integration space.







probability-theory probability-distributions uniform-distribution






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share|cite|improve this question













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edited Dec 4 '18 at 15:24









Did

247k23222458




247k23222458










asked Dec 4 '18 at 12:49









AromaTheLoopAromaTheLoop

464




464








  • 1




    $begingroup$
    (Big) Hint: By symmetry, this is equal to $$P(X_1+X_2+X_3+X_4le1)$$ Now, compute the latter...
    $endgroup$
    – Did
    Dec 4 '18 at 15:23










  • $begingroup$
    @Did I can't figure that out. Looks like it gets us to equation $$F(1) = 1 - F(3)$$ but I can't recall any formal symmetry connected with that,
    $endgroup$
    – AromaTheLoop
    Dec 4 '18 at 17:07






  • 1




    $begingroup$
    Sub-hint: $(1-X_1,1-X_2,1-X_3,1-X_4)$ is distributed like $(X_1,X_2,X_3,X_4)$.
    $endgroup$
    – Did
    Dec 4 '18 at 17:36










  • $begingroup$
    @Did Oh I see, thanks!
    $endgroup$
    – AromaTheLoop
    Dec 4 '18 at 17:54
















  • 1




    $begingroup$
    (Big) Hint: By symmetry, this is equal to $$P(X_1+X_2+X_3+X_4le1)$$ Now, compute the latter...
    $endgroup$
    – Did
    Dec 4 '18 at 15:23










  • $begingroup$
    @Did I can't figure that out. Looks like it gets us to equation $$F(1) = 1 - F(3)$$ but I can't recall any formal symmetry connected with that,
    $endgroup$
    – AromaTheLoop
    Dec 4 '18 at 17:07






  • 1




    $begingroup$
    Sub-hint: $(1-X_1,1-X_2,1-X_3,1-X_4)$ is distributed like $(X_1,X_2,X_3,X_4)$.
    $endgroup$
    – Did
    Dec 4 '18 at 17:36










  • $begingroup$
    @Did Oh I see, thanks!
    $endgroup$
    – AromaTheLoop
    Dec 4 '18 at 17:54










1




1




$begingroup$
(Big) Hint: By symmetry, this is equal to $$P(X_1+X_2+X_3+X_4le1)$$ Now, compute the latter...
$endgroup$
– Did
Dec 4 '18 at 15:23




$begingroup$
(Big) Hint: By symmetry, this is equal to $$P(X_1+X_2+X_3+X_4le1)$$ Now, compute the latter...
$endgroup$
– Did
Dec 4 '18 at 15:23












$begingroup$
@Did I can't figure that out. Looks like it gets us to equation $$F(1) = 1 - F(3)$$ but I can't recall any formal symmetry connected with that,
$endgroup$
– AromaTheLoop
Dec 4 '18 at 17:07




$begingroup$
@Did I can't figure that out. Looks like it gets us to equation $$F(1) = 1 - F(3)$$ but I can't recall any formal symmetry connected with that,
$endgroup$
– AromaTheLoop
Dec 4 '18 at 17:07




1




1




$begingroup$
Sub-hint: $(1-X_1,1-X_2,1-X_3,1-X_4)$ is distributed like $(X_1,X_2,X_3,X_4)$.
$endgroup$
– Did
Dec 4 '18 at 17:36




$begingroup$
Sub-hint: $(1-X_1,1-X_2,1-X_3,1-X_4)$ is distributed like $(X_1,X_2,X_3,X_4)$.
$endgroup$
– Did
Dec 4 '18 at 17:36












$begingroup$
@Did Oh I see, thanks!
$endgroup$
– AromaTheLoop
Dec 4 '18 at 17:54






$begingroup$
@Did Oh I see, thanks!
$endgroup$
– AromaTheLoop
Dec 4 '18 at 17:54












1 Answer
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Once we get to the step proposed by @Did, we can obtain the solution easily using geometric probability. Our probability here would be the hyper-volume covered by
$$x_1+x_2+x_3+x_4 leq 1 text{ and } 0leq x_i leq 1$$ is exactly the hyper-volume of an $4$-dimensional simplex, which is $dfrac{1}{4!} = dfrac{1}{24}$.






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    $begingroup$

    Once we get to the step proposed by @Did, we can obtain the solution easily using geometric probability. Our probability here would be the hyper-volume covered by
    $$x_1+x_2+x_3+x_4 leq 1 text{ and } 0leq x_i leq 1$$ is exactly the hyper-volume of an $4$-dimensional simplex, which is $dfrac{1}{4!} = dfrac{1}{24}$.






    share|cite|improve this answer









    $endgroup$


















      0












      $begingroup$

      Once we get to the step proposed by @Did, we can obtain the solution easily using geometric probability. Our probability here would be the hyper-volume covered by
      $$x_1+x_2+x_3+x_4 leq 1 text{ and } 0leq x_i leq 1$$ is exactly the hyper-volume of an $4$-dimensional simplex, which is $dfrac{1}{4!} = dfrac{1}{24}$.






      share|cite|improve this answer









      $endgroup$
















        0












        0








        0





        $begingroup$

        Once we get to the step proposed by @Did, we can obtain the solution easily using geometric probability. Our probability here would be the hyper-volume covered by
        $$x_1+x_2+x_3+x_4 leq 1 text{ and } 0leq x_i leq 1$$ is exactly the hyper-volume of an $4$-dimensional simplex, which is $dfrac{1}{4!} = dfrac{1}{24}$.






        share|cite|improve this answer









        $endgroup$



        Once we get to the step proposed by @Did, we can obtain the solution easily using geometric probability. Our probability here would be the hyper-volume covered by
        $$x_1+x_2+x_3+x_4 leq 1 text{ and } 0leq x_i leq 1$$ is exactly the hyper-volume of an $4$-dimensional simplex, which is $dfrac{1}{4!} = dfrac{1}{24}$.







        share|cite|improve this answer












        share|cite|improve this answer



        share|cite|improve this answer










        answered Dec 4 '18 at 15:40









        Isaac BrowneIsaac Browne

        4,59731132




        4,59731132






























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