Computing $mathbbPX_(1)+X_(2) > X_(3)$ where $X_1,X_2,X_3$ are i.i.d $U(0,1)$Find $P(x_2/x_3 leq a)$ where $x_1,x_2,dots,x_n $ are iid $operatornameUniform(0,1)$For exponential random variables $X_i$, how to find $P(t-X_1<X_2mid t-X_1<X_3)$?Simply question about i.i.d uniform random variables on (0,1)Probability and expectation of three ordered random variablesGiving an example of a t-distribution with independent $X_1, X_2, X_3$Finding $P(X_1+X_2 > 1.9X_3)$ where $X_1$, $X_2$, and $X_3$ are independent, normal distributed random variablesHow can I compute an expression for $P(X_1>X_2>X_3>X_4)$ if $X_1,X_2,X_3,X_4$ are normal and mututally independent?Finding $P(X_1+X_2+X_3+X_4ge3)$ for independent $X_isim U(0,1)$Find $P(X_1 > max(X_2, X_3)| X_1, X_2, X_3 geq d)$Compute $P(X_(2) ≤ 3X_(1))$ by using the integration technique

Strange opamp's output impedance in spice

"If + would" conditional in present perfect tense

Locked Away- What am I?

Was it really inappropriate to write a pull request for the company I interviewed with?

How to make sure I'm assertive enough in contact with subordinates?

Under what conditions can the right to remain silent be revoked in the USA?

Computation logic of Partway in TikZ

Rationale to prefer local variables over instance variables?

Is it possible to clone a polymorphic object without manually adding overridden clone method into each derived class in C++?

How do spaceships determine each other's mass in space?

What does *dead* mean in *What do you mean, dead?*?

How can a demon take control of a human body during REM sleep?

Is there a logarithm base for which the logarithm becomes an identity function?

Will expression retain the same definition if particle is changed?

Do Paladin Auras of Differing Oaths Stack?

Why do we say 'Pairwise Disjoint', rather than 'Disjoint'?

Smooth vector fields on a surface modulo diffeomorphisms

Are these two graphs isomorphic? Why/Why not?

How do you make a gun that shoots melee weapons and/or swords?

What is better: yes / no radio, or simple checkbox?

Is there stress on two letters on the word стоят

Is it appropriate to ask a former professor to order a book for me through an inter-library loan?

What is this tube in a jet engine's air intake?

What would be the most expensive material to an intergalactic society?



Computing $mathbbPX_(1)+X_(2) > X_(3)$ where $X_1,X_2,X_3$ are i.i.d $U(0,1)$


Find $P(x_2/x_3 leq a)$ where $x_1,x_2,dots,x_n $ are iid $operatornameUniform(0,1)$For exponential random variables $X_i$, how to find $P(t-X_1<X_2mid t-X_1<X_3)$?Simply question about i.i.d uniform random variables on (0,1)Probability and expectation of three ordered random variablesGiving an example of a t-distribution with independent $X_1, X_2, X_3$Finding $P(X_1+X_2 > 1.9X_3)$ where $X_1$, $X_2$, and $X_3$ are independent, normal distributed random variablesHow can I compute an expression for $P(X_1>X_2>X_3>X_4)$ if $X_1,X_2,X_3,X_4$ are normal and mututally independent?Finding $P(X_1+X_2+X_3+X_4ge3)$ for independent $X_isim U(0,1)$Find $P(X_1 > max(X_2, X_3)| X_1, X_2, X_3 geq d)$Compute $P(X_(2) ≤ 3X_(1))$ by using the integration technique













1












$begingroup$



Suppose that $X_1 , X_2 , X_3$ are independent $U (0, 1)$-distributed random variables and let $(X_(1) , X_(2) , X _(3) )$ be the corresponding order statistic. Compute $mathbbPX_(1)+X_(2) > X_(3)$.




I have the joint of the distribution which is $3!$



Also, I have:



$$mathbbPX_(1)+X_(2) > X_(3)=1-mathbbPX_(1)+X_(2) le X_(3)$$ with $mathbb 0<x_1<minx_2,x_3-x_2, quad 0<x_2<x_3, quad0<x_3<1$



But when I do:



$$1-6left(int_0^1int_0^x_3int_0^x_2 ,dx_1,dx_2,dx_3 +int_0^1int_0^x_3int_0^x_3-x_2 ,dx_1,dx_2,dx_3right)=-3$$



I do not really know what I'm doing wrong.






probability probability-theory probability-distributions uniform-distribution order-statistics







share|cite|improve this question











$endgroup$
















    1












    $begingroup$



    Suppose that $X_1 , X_2 , X_3$ are independent $U (0, 1)$-distributed random variables and let $(X_(1) , X_(2) , X _(3) )$ be the corresponding order statistic. Compute $mathbbPX_(1)+X_(2) > X_(3)$.




    I have the joint of the distribution which is $3!$



    Also, I have:



    $$mathbbPX_(1)+X_(2) > X_(3)=1-mathbbPX_(1)+X_(2) le X_(3)$$ with $mathbb 0<x_1<minx_2,x_3-x_2, quad 0<x_2<x_3, quad0<x_3<1$



    But when I do:



    $$1-6left(int_0^1int_0^x_3int_0^x_2 ,dx_1,dx_2,dx_3 +int_0^1int_0^x_3int_0^x_3-x_2 ,dx_1,dx_2,dx_3right)=-3$$



    I do not really know what I'm doing wrong.






    probability probability-theory probability-distributions uniform-distribution order-statistics







    share|cite|improve this question











    $endgroup$














      1












      1








      1


      0



      $begingroup$



      Suppose that $X_1 , X_2 , X_3$ are independent $U (0, 1)$-distributed random variables and let $(X_(1) , X_(2) , X _(3) )$ be the corresponding order statistic. Compute $mathbbPX_(1)+X_(2) > X_(3)$.




      I have the joint of the distribution which is $3!$



      Also, I have:



      $$mathbbPX_(1)+X_(2) > X_(3)=1-mathbbPX_(1)+X_(2) le X_(3)$$ with $mathbb 0<x_1<minx_2,x_3-x_2, quad 0<x_2<x_3, quad0<x_3<1$



      But when I do:



      $$1-6left(int_0^1int_0^x_3int_0^x_2 ,dx_1,dx_2,dx_3 +int_0^1int_0^x_3int_0^x_3-x_2 ,dx_1,dx_2,dx_3right)=-3$$



      I do not really know what I'm doing wrong.






      probability probability-theory probability-distributions uniform-distribution order-statistics







      share|cite|improve this question











      $endgroup$





      Suppose that $X_1 , X_2 , X_3$ are independent $U (0, 1)$-distributed random variables and let $(X_(1) , X_(2) , X _(3) )$ be the corresponding order statistic. Compute $mathbbPX_(1)+X_(2) > X_(3)$.




      I have the joint of the distribution which is $3!$



      Also, I have:



      $$mathbbPX_(1)+X_(2) > X_(3)=1-mathbbPX_(1)+X_(2) le X_(3)$$ with $mathbb 0<x_1<minx_2,x_3-x_2, quad 0<x_2<x_3, quad0<x_3<1$



      But when I do:



      $$1-6left(int_0^1int_0^x_3int_0^x_2 ,dx_1,dx_2,dx_3 +int_0^1int_0^x_3int_0^x_3-x_2 ,dx_1,dx_2,dx_3right)=-3$$



      I do not really know what I'm doing wrong.






      probability probability-theory probability-distributions uniform-distribution order-statistics




      probability probability-theory probability-distributions uniform-distribution order-statistics






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited yesterday









      StubbornAtom

      6,14311339




      6,14311339










      asked yesterday









      Mahamad A. KanoutéMahamad A. Kanouté

      395113




      395113




















          1 Answer
          1






          active

          oldest

          votes


















          1












          $begingroup$

          There could possibly be a simpler argument that does not require doing the integration explicitly.



          For the integration, I felt it convenient to use a change of variables.



          You are looking for



          $$P(X_(1)+X_(2)<X_(3))=6iiintmathbf1_x+y<z,,,0<x<y<z<1,mathrmdx,mathrmdy,mathrmdz$$



          Change variables $(x,y,z)to(u,v,w)$ with $$u=x+y,,,v=y,,,w=z$$



          Then, $$x+y<z,,,0<x<y<z<1implies u<w,,,0<u-v<v<w<1$$



          Therefore,



          beginalign
          iiintmathbf1_x+y<z,,,0<x<y<z<1,mathrmdx,mathrmdy,mathrmdz&=int_0^1int_0^wint_v^min(2v,w),mathrmdu,mathrmdv,mathrmdw
          \&=int_0^1int_0^w/2int_v^2v,mathrmdu,mathrmdv,mathrmdw+int_0^1int_w/2^wint_v^w,mathrmdu,mathrmdv,mathrmdw
          \&=frac112
          endalign






          share|cite|improve this answer











          $endgroup$












          • $begingroup$
            why does v goes from 0 to w/2 ?
            $endgroup$
            – Mahamad A. Kanouté
            yesterday










          • $begingroup$
            I get -1 as answer
            $endgroup$
            – Mahamad A. Kanouté
            yesterday










          • $begingroup$
            @MahamadA.Kanouté See the breaking of the integral.
            $endgroup$
            – StubbornAtom
            22 hours ago










          Your Answer





          StackExchange.ifUsing("editor", function ()
          return StackExchange.using("mathjaxEditing", function ()
          StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix)
          StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
          );
          );
          , "mathjax-editing");

          StackExchange.ready(function()
          var channelOptions =
          tags: "".split(" "),
          id: "69"
          ;
          initTagRenderer("".split(" "), "".split(" "), channelOptions);

          StackExchange.using("externalEditor", function()
          // Have to fire editor after snippets, if snippets enabled
          if (StackExchange.settings.snippets.snippetsEnabled)
          StackExchange.using("snippets", function()
          createEditor();
          );

          else
          createEditor();

          );

          function createEditor()
          StackExchange.prepareEditor(
          heartbeatType: 'answer',
          autoActivateHeartbeat: false,
          convertImagesToLinks: true,
          noModals: true,
          showLowRepImageUploadWarning: true,
          reputationToPostImages: 10,
          bindNavPrevention: true,
          postfix: "",
          imageUploader:
          brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
          contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
          allowUrls: true
          ,
          noCode: true, onDemand: true,
          discardSelector: ".discard-answer"
          ,immediatelyShowMarkdownHelp:true
          );



          );













          draft saved

          draft discarded


















          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3140526%2fcomputing-mathbbp-x-1x-2-x-3-where-x-1-x-2-x-3-are-i-i%23new-answer', 'question_page');

          );

          Post as a guest















          Required, but never shown

























          1 Answer
          1






          active

          oldest

          votes








          1 Answer
          1






          active

          oldest

          votes









          active

          oldest

          votes






          active

          oldest

          votes









          1












          $begingroup$

          There could possibly be a simpler argument that does not require doing the integration explicitly.



          For the integration, I felt it convenient to use a change of variables.



          You are looking for



          $$P(X_(1)+X_(2)<X_(3))=6iiintmathbf1_x+y<z,,,0<x<y<z<1,mathrmdx,mathrmdy,mathrmdz$$



          Change variables $(x,y,z)to(u,v,w)$ with $$u=x+y,,,v=y,,,w=z$$



          Then, $$x+y<z,,,0<x<y<z<1implies u<w,,,0<u-v<v<w<1$$



          Therefore,



          beginalign
          iiintmathbf1_x+y<z,,,0<x<y<z<1,mathrmdx,mathrmdy,mathrmdz&=int_0^1int_0^wint_v^min(2v,w),mathrmdu,mathrmdv,mathrmdw
          \&=int_0^1int_0^w/2int_v^2v,mathrmdu,mathrmdv,mathrmdw+int_0^1int_w/2^wint_v^w,mathrmdu,mathrmdv,mathrmdw
          \&=frac112
          endalign






          share|cite|improve this answer











          $endgroup$












          • $begingroup$
            why does v goes from 0 to w/2 ?
            $endgroup$
            – Mahamad A. Kanouté
            yesterday










          • $begingroup$
            I get -1 as answer
            $endgroup$
            – Mahamad A. Kanouté
            yesterday










          • $begingroup$
            @MahamadA.Kanouté See the breaking of the integral.
            $endgroup$
            – StubbornAtom
            22 hours ago















          1












          $begingroup$

          There could possibly be a simpler argument that does not require doing the integration explicitly.



          For the integration, I felt it convenient to use a change of variables.



          You are looking for



          $$P(X_(1)+X_(2)<X_(3))=6iiintmathbf1_x+y<z,,,0<x<y<z<1,mathrmdx,mathrmdy,mathrmdz$$



          Change variables $(x,y,z)to(u,v,w)$ with $$u=x+y,,,v=y,,,w=z$$



          Then, $$x+y<z,,,0<x<y<z<1implies u<w,,,0<u-v<v<w<1$$



          Therefore,



          beginalign
          iiintmathbf1_x+y<z,,,0<x<y<z<1,mathrmdx,mathrmdy,mathrmdz&=int_0^1int_0^wint_v^min(2v,w),mathrmdu,mathrmdv,mathrmdw
          \&=int_0^1int_0^w/2int_v^2v,mathrmdu,mathrmdv,mathrmdw+int_0^1int_w/2^wint_v^w,mathrmdu,mathrmdv,mathrmdw
          \&=frac112
          endalign






          share|cite|improve this answer











          $endgroup$












          • $begingroup$
            why does v goes from 0 to w/2 ?
            $endgroup$
            – Mahamad A. Kanouté
            yesterday










          • $begingroup$
            I get -1 as answer
            $endgroup$
            – Mahamad A. Kanouté
            yesterday










          • $begingroup$
            @MahamadA.Kanouté See the breaking of the integral.
            $endgroup$
            – StubbornAtom
            22 hours ago













          1












          1








          1





          $begingroup$

          There could possibly be a simpler argument that does not require doing the integration explicitly.



          For the integration, I felt it convenient to use a change of variables.



          You are looking for



          $$P(X_(1)+X_(2)<X_(3))=6iiintmathbf1_x+y<z,,,0<x<y<z<1,mathrmdx,mathrmdy,mathrmdz$$



          Change variables $(x,y,z)to(u,v,w)$ with $$u=x+y,,,v=y,,,w=z$$



          Then, $$x+y<z,,,0<x<y<z<1implies u<w,,,0<u-v<v<w<1$$



          Therefore,



          beginalign
          iiintmathbf1_x+y<z,,,0<x<y<z<1,mathrmdx,mathrmdy,mathrmdz&=int_0^1int_0^wint_v^min(2v,w),mathrmdu,mathrmdv,mathrmdw
          \&=int_0^1int_0^w/2int_v^2v,mathrmdu,mathrmdv,mathrmdw+int_0^1int_w/2^wint_v^w,mathrmdu,mathrmdv,mathrmdw
          \&=frac112
          endalign






          share|cite|improve this answer











          $endgroup$



          There could possibly be a simpler argument that does not require doing the integration explicitly.



          For the integration, I felt it convenient to use a change of variables.



          You are looking for



          $$P(X_(1)+X_(2)<X_(3))=6iiintmathbf1_x+y<z,,,0<x<y<z<1,mathrmdx,mathrmdy,mathrmdz$$



          Change variables $(x,y,z)to(u,v,w)$ with $$u=x+y,,,v=y,,,w=z$$



          Then, $$x+y<z,,,0<x<y<z<1implies u<w,,,0<u-v<v<w<1$$



          Therefore,



          beginalign
          iiintmathbf1_x+y<z,,,0<x<y<z<1,mathrmdx,mathrmdy,mathrmdz&=int_0^1int_0^wint_v^min(2v,w),mathrmdu,mathrmdv,mathrmdw
          \&=int_0^1int_0^w/2int_v^2v,mathrmdu,mathrmdv,mathrmdw+int_0^1int_w/2^wint_v^w,mathrmdu,mathrmdv,mathrmdw
          \&=frac112
          endalign







          share|cite|improve this answer














          share|cite|improve this answer



          share|cite|improve this answer








          edited 22 hours ago

























          answered yesterday









          StubbornAtomStubbornAtom

          6,14311339




          6,14311339











          • $begingroup$
            why does v goes from 0 to w/2 ?
            $endgroup$
            – Mahamad A. Kanouté
            yesterday










          • $begingroup$
            I get -1 as answer
            $endgroup$
            – Mahamad A. Kanouté
            yesterday










          • $begingroup$
            @MahamadA.Kanouté See the breaking of the integral.
            $endgroup$
            – StubbornAtom
            22 hours ago
















          • $begingroup$
            why does v goes from 0 to w/2 ?
            $endgroup$
            – Mahamad A. Kanouté
            yesterday










          • $begingroup$
            I get -1 as answer
            $endgroup$
            – Mahamad A. Kanouté
            yesterday










          • $begingroup$
            @MahamadA.Kanouté See the breaking of the integral.
            $endgroup$
            – StubbornAtom
            22 hours ago















          $begingroup$
          why does v goes from 0 to w/2 ?
          $endgroup$
          – Mahamad A. Kanouté
          yesterday




          $begingroup$
          why does v goes from 0 to w/2 ?
          $endgroup$
          – Mahamad A. Kanouté
          yesterday












          $begingroup$
          I get -1 as answer
          $endgroup$
          – Mahamad A. Kanouté
          yesterday




          $begingroup$
          I get -1 as answer
          $endgroup$
          – Mahamad A. Kanouté
          yesterday












          $begingroup$
          @MahamadA.Kanouté See the breaking of the integral.
          $endgroup$
          – StubbornAtom
          22 hours ago




          $begingroup$
          @MahamadA.Kanouté See the breaking of the integral.
          $endgroup$
          – StubbornAtom
          22 hours ago

















          draft saved

          draft discarded
















































          Thanks for contributing an answer to Mathematics Stack Exchange!


          • Please be sure to answer the question. Provide details and share your research!

          But avoid …


          • Asking for help, clarification, or responding to other answers.

          • Making statements based on opinion; back them up with references or personal experience.

          Use MathJax to format equations. MathJax reference.


          To learn more, see our tips on writing great answers.




          draft saved


          draft discarded














          StackExchange.ready(
          function ()
          StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3140526%2fcomputing-mathbbp-x-1x-2-x-3-where-x-1-x-2-x-3-are-i-i%23new-answer', 'question_page');

          );

          Post as a guest















          Required, but never shown





















































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown

































          Required, but never shown














          Required, but never shown












          Required, but never shown







          Required, but never shown







          -

          Popular posts from this blog

          How should I support this large drywall patch? Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern) Announcing the arrival of Valued Associate #679: Cesar Manara Unicorn Meta Zoo #1: Why another podcast?How do I cover large gaps in drywall?How do I keep drywall around a patch from crumbling?Can I glue a second layer of drywall?How to patch long strip on drywall?Large drywall patch: how to avoid bulging seams?Drywall Mesh Patch vs. Bulge? To remove or not to remove?How to fix this drywall job?Prep drywall before backsplashWhat's the best way to fix this horrible drywall patch job?Drywall patching using 3M Patch Plus Primer

          Image
          How do living politicians protect their readily obtainable signatures from misuse? Is it possible to force a specific program to remain in memory after closing it? File name problem(?) How to pronounce 伝統色 How does the math work when buying airline miles? Sum letters are not two different An adverb for when you're not exaggerating What does it mean that physics no longer uses mechanical models to describe phenomena? Take 2! Is this homebrew Lady of Pain warlock patron balanced? How many time has Arya actually used Needle? A letter with no particular backstory Converted a Scalar function to a TVF function for parallel execution-Still running in Serial mode Generate an RGB colour grid Co-worker has annoying ringtone Strange behavior of Object.defineProperty() in JavaScript "Lost his faith in humanity in the trenches of Verdun" — last line of an SF story What is this round thing on the pantry door in The Shining What would you ca...
          Read more

          random experiment with two different functions on unit interval Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Random variable and probability space notionsRandom Walk with EdgesFinding functions where the increase over a random interval is Poisson distributedNumber of days until dayCan an observed event in fact be of zero probability?Unit random processmodels of coins and uniform distributionHow to get the number of successes given $n$ trials , probability $P$ and a random variable $X$Absorbing Markov chain in a computer. Is “almost every” turned into always convergence in computer executions?Stopped random walk is not uniformly integrable

          Image
          What would you call this weird metallic apparatus that allows you to lift people? Is it possible to give , in economics, an example of a relation ( set of ordered pairs) that is not a function? Maximum summed subsequences with non-adjacent items How many time has Arya actually used Needle? Is there public access to the Meteor Crater in Arizona? Belief In God or Knowledge Of God. Which is better? What is the meaning of 'breadth' in breadth first search? Why is there Net Work Done on a Pressure/Volume Cycle? How much damage would a cupful of neutron star matter do to the Earth? Sum letters are not two different Why does it sometimes sound good to play a grace note as a lead in to a note in a melody? What initially awakened the Balrog? How to pronounce 伝統色 QGIS virtual layer functionality does not seem to support memory layers AppleTVs create a chatty alternate WiFi network How to write capital alpha? How often does castling occur in gra...
          Read more

          Lowndes Grove History Architecture References Navigation menu32°48′6″N 79°57′58″W / 32.80167°N 79.96611°W / 32.80167; -79.9661132°48′6″N 79°57′58″W / 32.80167°N 79.96611°W / 32.80167; -79.9661178002500"National Register Information System"Historic houses of South Carolina"Lowndes Grove""+32° 48' 6.00", −79° 57' 58.00""Lowndes Grove, Charleston County (260 St. Margaret St., Charleston)""Lowndes Grove"The Charleston ExpositionIt Happened in South Carolina"Lowndes Grove (House), Saint Margaret Street & Sixth Avenue, Charleston, Charleston County, SC(Photographs)"Plantations of the Carolina Low Countrye

          Image
          Contributing propertyKeeper of the RegisterHistoric districtHistory of the National Register of Historic PlacesNational Park ServiceProperty typesCharlestonColumbiaGreenvilleNorth CharlestonRock Hill Houses on the National Register of Historic Places in South CarolinaGeorgian architecture in South CarolinaHouses completed in 1786Plantation houses in South CarolinaNational Register of Historic Places in Charleston, South CarolinaHouses in Charleston, South Carolina plantation houseAshley RiverCharlestonNational Register of Historic PlacesRevolutionary WarWilliam LowndesU.S. CongressSouth Carolina Inter-State and West Indian ExpositionGeorgianDoricbalustradespedimentoculusentablaturemodilliondentillightship roofterra cottatilescorbelledcornicesRegency stylespiral staircaseskylight Lowndes Grove U.S. National Register of Historic Places Lowndes Grove in 1940 Show map of South Carolina Show map of the United States Location 260 St. Margaret St., Charleston, South Carolina Coordinates 32...
          Read more