How to calculate $mathbb{P}(|U_{1} - U_{2}| < frac{1}{12})$, where $U_{i} sim mathcal{U}(0, frac{1}{2})$...












4












$begingroup$


We have a Poisson process of intensity $lambda = 4$. We have the following event:



$A$: "Two marks appear with a separation of $frac{1}{12}$ or less".



We need to calculate the probability of exactly two "marks" appearing between $0$ and $frac{1}{2}$, and $A$ ocurring at the same time. That is, the probability of exactly two marks ocurring in $(0, frac{1}{2})$ with a separation of $frac{1}{12}$ or less.



Here's how far I've gotten:



Let $N$: Number of marks between $0$ and $frac{1}{2}$. We want:
$$mathbb{P}(N=2, A) = mathbb{P}(A|N=2)mathbb{P}(N=2) $$



$Nsim Poi(frac{1}{2}4)$, so: $$mathbb{P}(N=2) = frac{2^2}{2!}e^{-2} = 2e^{-2}$$



The arrival times for a conditioned number of marks (in this case two) is $U_{1}$, and $U_{2}$, where $U_{i} sim mathcal{U}(0, frac{1}{2})$, and $U_{i}$ are independent, so:



$$mathbb{P}(A|N=2) = mathbb{P}(max(U_{1}, U_{2}) - min(U_{1}, U_{2}) < frac{1}{12}) = mathbb{P}left(|U_{1} - U_{2}| < frac{1}{12}right)$$



This is where I'm stuck. The answer is supposed to be $frac{11}{18}e^{-2}$, so $mathbb{P}left(|U_{1} - U_{2}| < frac{1}{12}right)$ should be $frac{11}{36}$.



Thanks.






probability probability-theory probability-distributions poisson-distribution







share|cite|improve this question











$endgroup$

















    4












    $begingroup$


    We have a Poisson process of intensity $lambda = 4$. We have the following event:



    $A$: "Two marks appear with a separation of $frac{1}{12}$ or less".



    We need to calculate the probability of exactly two "marks" appearing between $0$ and $frac{1}{2}$, and $A$ ocurring at the same time. That is, the probability of exactly two marks ocurring in $(0, frac{1}{2})$ with a separation of $frac{1}{12}$ or less.



    Here's how far I've gotten:



    Let $N$: Number of marks between $0$ and $frac{1}{2}$. We want:
    $$mathbb{P}(N=2, A) = mathbb{P}(A|N=2)mathbb{P}(N=2) $$



    $Nsim Poi(frac{1}{2}4)$, so: $$mathbb{P}(N=2) = frac{2^2}{2!}e^{-2} = 2e^{-2}$$



    The arrival times for a conditioned number of marks (in this case two) is $U_{1}$, and $U_{2}$, where $U_{i} sim mathcal{U}(0, frac{1}{2})$, and $U_{i}$ are independent, so:



    $$mathbb{P}(A|N=2) = mathbb{P}(max(U_{1}, U_{2}) - min(U_{1}, U_{2}) < frac{1}{12}) = mathbb{P}left(|U_{1} - U_{2}| < frac{1}{12}right)$$



    This is where I'm stuck. The answer is supposed to be $frac{11}{18}e^{-2}$, so $mathbb{P}left(|U_{1} - U_{2}| < frac{1}{12}right)$ should be $frac{11}{36}$.



    Thanks.






    probability probability-theory probability-distributions poisson-distribution







    share|cite|improve this question











    $endgroup$















      4












      4








      4


      0



      $begingroup$


      We have a Poisson process of intensity $lambda = 4$. We have the following event:



      $A$: "Two marks appear with a separation of $frac{1}{12}$ or less".



      We need to calculate the probability of exactly two "marks" appearing between $0$ and $frac{1}{2}$, and $A$ ocurring at the same time. That is, the probability of exactly two marks ocurring in $(0, frac{1}{2})$ with a separation of $frac{1}{12}$ or less.



      Here's how far I've gotten:



      Let $N$: Number of marks between $0$ and $frac{1}{2}$. We want:
      $$mathbb{P}(N=2, A) = mathbb{P}(A|N=2)mathbb{P}(N=2) $$



      $Nsim Poi(frac{1}{2}4)$, so: $$mathbb{P}(N=2) = frac{2^2}{2!}e^{-2} = 2e^{-2}$$



      The arrival times for a conditioned number of marks (in this case two) is $U_{1}$, and $U_{2}$, where $U_{i} sim mathcal{U}(0, frac{1}{2})$, and $U_{i}$ are independent, so:



      $$mathbb{P}(A|N=2) = mathbb{P}(max(U_{1}, U_{2}) - min(U_{1}, U_{2}) < frac{1}{12}) = mathbb{P}left(|U_{1} - U_{2}| < frac{1}{12}right)$$



      This is where I'm stuck. The answer is supposed to be $frac{11}{18}e^{-2}$, so $mathbb{P}left(|U_{1} - U_{2}| < frac{1}{12}right)$ should be $frac{11}{36}$.



      Thanks.






      probability probability-theory probability-distributions poisson-distribution







      share|cite|improve this question











      $endgroup$




      We have a Poisson process of intensity $lambda = 4$. We have the following event:



      $A$: "Two marks appear with a separation of $frac{1}{12}$ or less".



      We need to calculate the probability of exactly two "marks" appearing between $0$ and $frac{1}{2}$, and $A$ ocurring at the same time. That is, the probability of exactly two marks ocurring in $(0, frac{1}{2})$ with a separation of $frac{1}{12}$ or less.



      Here's how far I've gotten:



      Let $N$: Number of marks between $0$ and $frac{1}{2}$. We want:
      $$mathbb{P}(N=2, A) = mathbb{P}(A|N=2)mathbb{P}(N=2) $$



      $Nsim Poi(frac{1}{2}4)$, so: $$mathbb{P}(N=2) = frac{2^2}{2!}e^{-2} = 2e^{-2}$$



      The arrival times for a conditioned number of marks (in this case two) is $U_{1}$, and $U_{2}$, where $U_{i} sim mathcal{U}(0, frac{1}{2})$, and $U_{i}$ are independent, so:



      $$mathbb{P}(A|N=2) = mathbb{P}(max(U_{1}, U_{2}) - min(U_{1}, U_{2}) < frac{1}{12}) = mathbb{P}left(|U_{1} - U_{2}| < frac{1}{12}right)$$



      This is where I'm stuck. The answer is supposed to be $frac{11}{18}e^{-2}$, so $mathbb{P}left(|U_{1} - U_{2}| < frac{1}{12}right)$ should be $frac{11}{36}$.



      Thanks.






      probability probability-theory probability-distributions poisson-distribution




      probability probability-theory probability-distributions poisson-distribution






      share|cite|improve this question















      share|cite|improve this question













      share|cite|improve this question




      share|cite|improve this question








      edited Dec 12 '18 at 10:10









      Davide Giraudo

      127k16151264




      127k16151264










      asked Dec 11 '18 at 22:00









      Juanma EloyJuanma Eloy

      5611516




      5611516






















          2 Answers
          2






          active

          oldest

          votes


















          3












          $begingroup$

          Hint: saying that $U_1,U_2$ are independent uniform $[0,1/2]$ is equivalent to saying that $(U_1,U_2)$ is uniformly distributed over the square $[0,1/2]times [0,1/2]={(x,y):0le xle 1/2,0le yle 1/2}$. Draw this square, draw the region where $|x-y|le frac1{12}$, and compute the fraction of this area over the total area.






          share|cite|improve this answer









          $endgroup$





















            2












            $begingroup$

            You can proceed e.g. as follows: Let $V_1=2U_1, V_2=2U_2$. Then $Z = |V_1-V_2|$ follows a triangular distribution with parameters $(a,b,c)=(0,1,0)$ , and you want
            $$
            mathbb{P}left{ |U_1-U_2| < frac{1}{12}right}
            = mathbb{P}left{ Z < frac{1}{6}right} = int_0^{1/6} f_Z(z)dz = int_0^{1/6} (2-2z)dz = boxed{frac{11}{36}}
            $$
            as claimed.






            share|cite|improve this answer











            $endgroup$













              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%2f3035890%2fhow-to-calculate-mathbbpu-1-u-2-frac112-where-u-i-sim%23new-answer', 'question_page');
              }
              );

              Post as a guest















              Required, but never shown

























              2 Answers
              2






              active

              oldest

              votes








              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              3












              $begingroup$

              Hint: saying that $U_1,U_2$ are independent uniform $[0,1/2]$ is equivalent to saying that $(U_1,U_2)$ is uniformly distributed over the square $[0,1/2]times [0,1/2]={(x,y):0le xle 1/2,0le yle 1/2}$. Draw this square, draw the region where $|x-y|le frac1{12}$, and compute the fraction of this area over the total area.






              share|cite|improve this answer









              $endgroup$


















                3












                $begingroup$

                Hint: saying that $U_1,U_2$ are independent uniform $[0,1/2]$ is equivalent to saying that $(U_1,U_2)$ is uniformly distributed over the square $[0,1/2]times [0,1/2]={(x,y):0le xle 1/2,0le yle 1/2}$. Draw this square, draw the region where $|x-y|le frac1{12}$, and compute the fraction of this area over the total area.






                share|cite|improve this answer









                $endgroup$
















                  3












                  3








                  3





                  $begingroup$

                  Hint: saying that $U_1,U_2$ are independent uniform $[0,1/2]$ is equivalent to saying that $(U_1,U_2)$ is uniformly distributed over the square $[0,1/2]times [0,1/2]={(x,y):0le xle 1/2,0le yle 1/2}$. Draw this square, draw the region where $|x-y|le frac1{12}$, and compute the fraction of this area over the total area.






                  share|cite|improve this answer









                  $endgroup$



                  Hint: saying that $U_1,U_2$ are independent uniform $[0,1/2]$ is equivalent to saying that $(U_1,U_2)$ is uniformly distributed over the square $[0,1/2]times [0,1/2]={(x,y):0le xle 1/2,0le yle 1/2}$. Draw this square, draw the region where $|x-y|le frac1{12}$, and compute the fraction of this area over the total area.







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered Dec 11 '18 at 22:36









                  Mike EarnestMike Earnest

                  23.4k12051




                  23.4k12051























                      2












                      $begingroup$

                      You can proceed e.g. as follows: Let $V_1=2U_1, V_2=2U_2$. Then $Z = |V_1-V_2|$ follows a triangular distribution with parameters $(a,b,c)=(0,1,0)$ , and you want
                      $$
                      mathbb{P}left{ |U_1-U_2| < frac{1}{12}right}
                      = mathbb{P}left{ Z < frac{1}{6}right} = int_0^{1/6} f_Z(z)dz = int_0^{1/6} (2-2z)dz = boxed{frac{11}{36}}
                      $$
                      as claimed.






                      share|cite|improve this answer











                      $endgroup$


















                        2












                        $begingroup$

                        You can proceed e.g. as follows: Let $V_1=2U_1, V_2=2U_2$. Then $Z = |V_1-V_2|$ follows a triangular distribution with parameters $(a,b,c)=(0,1,0)$ , and you want
                        $$
                        mathbb{P}left{ |U_1-U_2| < frac{1}{12}right}
                        = mathbb{P}left{ Z < frac{1}{6}right} = int_0^{1/6} f_Z(z)dz = int_0^{1/6} (2-2z)dz = boxed{frac{11}{36}}
                        $$
                        as claimed.






                        share|cite|improve this answer











                        $endgroup$
















                          2












                          2








                          2





                          $begingroup$

                          You can proceed e.g. as follows: Let $V_1=2U_1, V_2=2U_2$. Then $Z = |V_1-V_2|$ follows a triangular distribution with parameters $(a,b,c)=(0,1,0)$ , and you want
                          $$
                          mathbb{P}left{ |U_1-U_2| < frac{1}{12}right}
                          = mathbb{P}left{ Z < frac{1}{6}right} = int_0^{1/6} f_Z(z)dz = int_0^{1/6} (2-2z)dz = boxed{frac{11}{36}}
                          $$
                          as claimed.






                          share|cite|improve this answer











                          $endgroup$



                          You can proceed e.g. as follows: Let $V_1=2U_1, V_2=2U_2$. Then $Z = |V_1-V_2|$ follows a triangular distribution with parameters $(a,b,c)=(0,1,0)$ , and you want
                          $$
                          mathbb{P}left{ |U_1-U_2| < frac{1}{12}right}
                          = mathbb{P}left{ Z < frac{1}{6}right} = int_0^{1/6} f_Z(z)dz = int_0^{1/6} (2-2z)dz = boxed{frac{11}{36}}
                          $$
                          as claimed.







                          share|cite|improve this answer














                          share|cite|improve this answer



                          share|cite|improve this answer








                          edited Dec 12 '18 at 11:51









                          StubbornAtom

                          6,04811239




                          6,04811239










                          answered Dec 11 '18 at 22:39









                          Clement C.Clement C.

                          50.5k33891




                          50.5k33891






























                              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%2f3035890%2fhow-to-calculate-mathbbpu-1-u-2-frac112-where-u-i-sim%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

                              Willebadessen

                              Image
                              Wappen Deutschlandkarte 51.633055555556 9.0330555555556 295 Koordinaten: 51° 38′  N , 9° 2′  O Basisdaten Bundesland: Nordrhein-Westfalen Regierungsbezirk: Detmold Kreis: Höxter Höhe: 295 m ü. NHN Fläche: 128,41 km 2 Einwohner: 8227 (31. Dez. 2017) [1] Bevölkerungsdichte: 64 Einwohner je km 2 Postleitzahlen: 34435, 34437, 34439 Vorwahlen: 05644, 05646, 05642 Kfz-Kennzeichen: HX Gemeindeschlüssel: 05 7 62 040 Stadtgliederung: 13 Stadtteile Adresse der Stadtverwaltung: Abdinghofweg 1 34439 Willebadessen Website: www.willebadessen.de Bürgermeister: Hans Hermann Bluhm (CDU) Lage der Stadt Willebadessen im Kreis Höxter Willebadessen ist eine Kleinstadt in Nordrhein-Westfalen und gehört zum Kreis Höxter im Regierungsbezirk Detmold. Die Stadt liegt im Osten des Regierungsbezirkes mitten im Naturpark Teutoburger Wald/Eggegebirge. Für den heu...
                              Read more

                              Ida-Boy-Ed-Garten

                              Image
                              Die Lage des Ida-Boy-Ed-Gartens, rot markiert Der Ida-Boy-Ed-Garten Der Ida-Boy-Ed-Garten, Blick von der Wakenitzmauer durch das Tor in der Stadtmauer Der Ida-Boy-Ed-Garten ist eine Straße der Lübecker Altstadt. Inhaltsverzeichnis 1 Lage 2 Geschichte 3 Literatur 4 Weblinks Lage | Der etwa 90 Meter lange Ida-Boy-Ed-Garten befindet sich an der nördlichen Spitze der Altstadtinsel, vor dem Burgtor. Die in einem Bogen durch eine Grünanlage hangaufwärts führende Straße verbindet die Wakenitzmauer mit der Burgtorbrücke. Geschichte | Der heutige Ida-Boy-Ed-Garten wurde ursprünglich als Verlängerung der Wakenitzmauer angelegt. 1903 wurde in den erhaltenen Teil der mittelalterlichen Stadtmauer, an dem die Wakenitzmauer bei der Einmündung der Kaiserstraße begann, ein Tor mit einem großen Bogen für die Durchführung der Straße und zwei kleineren seitlichen Öffnungen für Fußgänger gebrochen, um für die neu errichteten Wohnhäuser in diesem Bereich ...
                              Read more

                              Residenzschloss Arolsen

                              Image
                              Residenzschloss Arolsen Residenzschloss Arolsen Entstehungszeit: 1710–1728 Erhaltungszustand: Vollständig Ständische Stellung: Grafen Ort: Bad Arolsen Geographische Lage 51° 22′ 51″  N , 9° 1′ 19″  O 51.380833333333 9.0219444444444 293 Koordinaten: 51° 22′ 51″  N , 9° 1′ 19″  O Höhe: 293  m ü. NHN Das Residenzschloss Arolsen ist ein barockes Schloss in Bad Arolsen im Kreis Waldeck-Frankenberg in Nordhessen (Deutschland). Das Schloss wurde als dreiflügelige Anlage gebaut, an die sich ein englischer Garten anschließt. Zentrales landschaftsarchitektonisches Gestaltungselement ist ein ausgedehntes Rondell. Inhaltsverzeichnis 1 Geschichte 1.1 Vorgängerbau 1.2 Bau und Geschichte des neuen Schlosses 1.3 Bibliothek 1.4 Finanzielle und staatsrechtliche Konsequenzen 1.5 Heutige Nutzung 2 Innenausstattung 3 Umgebung 3.1 Park 3.2 Wirtschaftsgebäude 4 Literatur 5 Weblinks 6...
                              Read more