How is the subtraction of a uniform (0, k) and its entire part distributed?












0












$begingroup$


Let X be a random variable distributed as $U[0, K]$ for an integer K. Find the density function of $Y = f (x) = x- [x]$, where [x] denotes the integer part of the real number x.



I think that [X] represents a discrete uniform but this by definition would be with values at x = 1,2, ... k, and its density would be 1/k which is the same as that of the continuous uniform.



Now, I understand that the distribution [X] is dependent on X, then I could not assume independence to use some kind of transformation because I do not know the joint.



I appreciate if you can give me some other way that I have not considered, thank you very much.










share|cite|improve this question









$endgroup$

















    0












    $begingroup$


    Let X be a random variable distributed as $U[0, K]$ for an integer K. Find the density function of $Y = f (x) = x- [x]$, where [x] denotes the integer part of the real number x.



    I think that [X] represents a discrete uniform but this by definition would be with values at x = 1,2, ... k, and its density would be 1/k which is the same as that of the continuous uniform.



    Now, I understand that the distribution [X] is dependent on X, then I could not assume independence to use some kind of transformation because I do not know the joint.



    I appreciate if you can give me some other way that I have not considered, thank you very much.










    share|cite|improve this question









    $endgroup$















      0












      0








      0





      $begingroup$


      Let X be a random variable distributed as $U[0, K]$ for an integer K. Find the density function of $Y = f (x) = x- [x]$, where [x] denotes the integer part of the real number x.



      I think that [X] represents a discrete uniform but this by definition would be with values at x = 1,2, ... k, and its density would be 1/k which is the same as that of the continuous uniform.



      Now, I understand that the distribution [X] is dependent on X, then I could not assume independence to use some kind of transformation because I do not know the joint.



      I appreciate if you can give me some other way that I have not considered, thank you very much.










      share|cite|improve this question









      $endgroup$




      Let X be a random variable distributed as $U[0, K]$ for an integer K. Find the density function of $Y = f (x) = x- [x]$, where [x] denotes the integer part of the real number x.



      I think that [X] represents a discrete uniform but this by definition would be with values at x = 1,2, ... k, and its density would be 1/k which is the same as that of the continuous uniform.



      Now, I understand that the distribution [X] is dependent on X, then I could not assume independence to use some kind of transformation because I do not know the joint.



      I appreciate if you can give me some other way that I have not considered, thank you very much.







      statistics uniform-distribution






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Dec 30 '18 at 23:16









      Cristian PerdomoCristian Perdomo

      61




      61






















          1 Answer
          1






          active

          oldest

          votes


















          0












          $begingroup$

          So, set ${x}=x-lfloor xrfloor$ to be the fractional part of $x$. Check that ${x} in [0,1)$. Hence, $f_X(x)$ is defined for $xin [0,1)$. Let's compute the CDF. Fix a $cin[0,1)$, and study $mathbb{P}({x}leq c)$. Observe that,
          $$
          {{x} leq c}=bigcup_{k=0}^{K-1}{xin [k,k+c)},
          $$

          hence $mathbb{P}({x}leq c)=Kcdot frac{c}{K}=c$. Hence, it turns out that, ${x}$ is uniform on $[0,1)$ (and also, $lfloor xrfloor$ is uniform on ${0,dots,K-1}$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks, I understood.
            $endgroup$
            – Cristian Perdomo
            Dec 31 '18 at 0:26











          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%2f3057274%2fhow-is-the-subtraction-of-a-uniform-0-k-and-its-entire-part-distributed%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









          0












          $begingroup$

          So, set ${x}=x-lfloor xrfloor$ to be the fractional part of $x$. Check that ${x} in [0,1)$. Hence, $f_X(x)$ is defined for $xin [0,1)$. Let's compute the CDF. Fix a $cin[0,1)$, and study $mathbb{P}({x}leq c)$. Observe that,
          $$
          {{x} leq c}=bigcup_{k=0}^{K-1}{xin [k,k+c)},
          $$

          hence $mathbb{P}({x}leq c)=Kcdot frac{c}{K}=c$. Hence, it turns out that, ${x}$ is uniform on $[0,1)$ (and also, $lfloor xrfloor$ is uniform on ${0,dots,K-1}$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks, I understood.
            $endgroup$
            – Cristian Perdomo
            Dec 31 '18 at 0:26
















          0












          $begingroup$

          So, set ${x}=x-lfloor xrfloor$ to be the fractional part of $x$. Check that ${x} in [0,1)$. Hence, $f_X(x)$ is defined for $xin [0,1)$. Let's compute the CDF. Fix a $cin[0,1)$, and study $mathbb{P}({x}leq c)$. Observe that,
          $$
          {{x} leq c}=bigcup_{k=0}^{K-1}{xin [k,k+c)},
          $$

          hence $mathbb{P}({x}leq c)=Kcdot frac{c}{K}=c$. Hence, it turns out that, ${x}$ is uniform on $[0,1)$ (and also, $lfloor xrfloor$ is uniform on ${0,dots,K-1}$.






          share|cite|improve this answer









          $endgroup$













          • $begingroup$
            Thanks, I understood.
            $endgroup$
            – Cristian Perdomo
            Dec 31 '18 at 0:26














          0












          0








          0





          $begingroup$

          So, set ${x}=x-lfloor xrfloor$ to be the fractional part of $x$. Check that ${x} in [0,1)$. Hence, $f_X(x)$ is defined for $xin [0,1)$. Let's compute the CDF. Fix a $cin[0,1)$, and study $mathbb{P}({x}leq c)$. Observe that,
          $$
          {{x} leq c}=bigcup_{k=0}^{K-1}{xin [k,k+c)},
          $$

          hence $mathbb{P}({x}leq c)=Kcdot frac{c}{K}=c$. Hence, it turns out that, ${x}$ is uniform on $[0,1)$ (and also, $lfloor xrfloor$ is uniform on ${0,dots,K-1}$.






          share|cite|improve this answer









          $endgroup$



          So, set ${x}=x-lfloor xrfloor$ to be the fractional part of $x$. Check that ${x} in [0,1)$. Hence, $f_X(x)$ is defined for $xin [0,1)$. Let's compute the CDF. Fix a $cin[0,1)$, and study $mathbb{P}({x}leq c)$. Observe that,
          $$
          {{x} leq c}=bigcup_{k=0}^{K-1}{xin [k,k+c)},
          $$

          hence $mathbb{P}({x}leq c)=Kcdot frac{c}{K}=c$. Hence, it turns out that, ${x}$ is uniform on $[0,1)$ (and also, $lfloor xrfloor$ is uniform on ${0,dots,K-1}$.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Dec 30 '18 at 23:21









          AaronAaron

          1,947415




          1,947415












          • $begingroup$
            Thanks, I understood.
            $endgroup$
            – Cristian Perdomo
            Dec 31 '18 at 0:26


















          • $begingroup$
            Thanks, I understood.
            $endgroup$
            – Cristian Perdomo
            Dec 31 '18 at 0:26
















          $begingroup$
          Thanks, I understood.
          $endgroup$
          – Cristian Perdomo
          Dec 31 '18 at 0:26




          $begingroup$
          Thanks, I understood.
          $endgroup$
          – Cristian Perdomo
          Dec 31 '18 at 0:26


















          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%2f3057274%2fhow-is-the-subtraction-of-a-uniform-0-k-and-its-entire-part-distributed%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

          Probability when a professor distributes a quiz and homework assignment to a class of n students.

          Aardman Animations

          Are they similar matrix