Determining whether two events are “the same” or not











up vote
1
down vote

favorite












I was studying probability theory using the textbook Introduction to Probability (2e) - Blitzstein & Hwang. I came across an example problem from a chapter regarding join distributions and had a question regarding a step in the solution. Here's the specific example question that I'm referring to:






Suppoe a chicken lays a random number of eggs, $N$ where $N$ follows Pois($lambda$). Each egg independently hatches with probability $p$. Let $X$ be the number of eggs that hatch and $Y$ be the number of eggs that fail to hatch. What is the joint PMF of $X$ and $Y$?






Textbook Solution (paraphrased):



Because the number of eggs depends on $N$, the total number of eggs laid, we must condition on $N$.



$$P(X=i, Y=j) = sum_{n = 0}^infty P(X=i, Y=j | N=n)P(N=n)$$



Among the possible values of $n$, the only one that is relevant is when $n=i + j$ due to the fact that any other case would not be consistent with the number of eggs hatched and the number of eggs that failed to hatch, since $X+Y$ must equal $N$. Therefore, we can drop all of the other terms in the summation, which leaves us with,



$$P(X=i, Y=j) = P(X=i, Y=j | N=i + j)P(N=i+j)$$



Conditional on $N=i+j$, the events $X=i$ and $Y=j$ are exactly the same event. We can therefore drop either $X$ or $Y$ and find the joint PMF with,



$$P(X=i, Y=j) = P(X=i, | N=i+j)P(N=i+j)$$






I omitted the remaining calculations as they are irrelevant to my question. The sentence that I put in bold print is the particular sentence that is confusing me.



How are $X$ and $Y$ - the number of eggs that hatch and don't hatch, respectively - the same events?



My initial understanding would be that due to the strong dependent relationship they have (i.e. knowing the value of one random variable automatically gives us the value of the other), we are able to reach this conclusion.



Is my assumption correct? Perhaps the way that the sentence was worded is confusing me.






Any feedback is appreciated. Thank you.










share|cite|improve this question


























    up vote
    1
    down vote

    favorite












    I was studying probability theory using the textbook Introduction to Probability (2e) - Blitzstein & Hwang. I came across an example problem from a chapter regarding join distributions and had a question regarding a step in the solution. Here's the specific example question that I'm referring to:






    Suppoe a chicken lays a random number of eggs, $N$ where $N$ follows Pois($lambda$). Each egg independently hatches with probability $p$. Let $X$ be the number of eggs that hatch and $Y$ be the number of eggs that fail to hatch. What is the joint PMF of $X$ and $Y$?






    Textbook Solution (paraphrased):



    Because the number of eggs depends on $N$, the total number of eggs laid, we must condition on $N$.



    $$P(X=i, Y=j) = sum_{n = 0}^infty P(X=i, Y=j | N=n)P(N=n)$$



    Among the possible values of $n$, the only one that is relevant is when $n=i + j$ due to the fact that any other case would not be consistent with the number of eggs hatched and the number of eggs that failed to hatch, since $X+Y$ must equal $N$. Therefore, we can drop all of the other terms in the summation, which leaves us with,



    $$P(X=i, Y=j) = P(X=i, Y=j | N=i + j)P(N=i+j)$$



    Conditional on $N=i+j$, the events $X=i$ and $Y=j$ are exactly the same event. We can therefore drop either $X$ or $Y$ and find the joint PMF with,



    $$P(X=i, Y=j) = P(X=i, | N=i+j)P(N=i+j)$$






    I omitted the remaining calculations as they are irrelevant to my question. The sentence that I put in bold print is the particular sentence that is confusing me.



    How are $X$ and $Y$ - the number of eggs that hatch and don't hatch, respectively - the same events?



    My initial understanding would be that due to the strong dependent relationship they have (i.e. knowing the value of one random variable automatically gives us the value of the other), we are able to reach this conclusion.



    Is my assumption correct? Perhaps the way that the sentence was worded is confusing me.






    Any feedback is appreciated. Thank you.










    share|cite|improve this question
























      up vote
      1
      down vote

      favorite









      up vote
      1
      down vote

      favorite











      I was studying probability theory using the textbook Introduction to Probability (2e) - Blitzstein & Hwang. I came across an example problem from a chapter regarding join distributions and had a question regarding a step in the solution. Here's the specific example question that I'm referring to:






      Suppoe a chicken lays a random number of eggs, $N$ where $N$ follows Pois($lambda$). Each egg independently hatches with probability $p$. Let $X$ be the number of eggs that hatch and $Y$ be the number of eggs that fail to hatch. What is the joint PMF of $X$ and $Y$?






      Textbook Solution (paraphrased):



      Because the number of eggs depends on $N$, the total number of eggs laid, we must condition on $N$.



      $$P(X=i, Y=j) = sum_{n = 0}^infty P(X=i, Y=j | N=n)P(N=n)$$



      Among the possible values of $n$, the only one that is relevant is when $n=i + j$ due to the fact that any other case would not be consistent with the number of eggs hatched and the number of eggs that failed to hatch, since $X+Y$ must equal $N$. Therefore, we can drop all of the other terms in the summation, which leaves us with,



      $$P(X=i, Y=j) = P(X=i, Y=j | N=i + j)P(N=i+j)$$



      Conditional on $N=i+j$, the events $X=i$ and $Y=j$ are exactly the same event. We can therefore drop either $X$ or $Y$ and find the joint PMF with,



      $$P(X=i, Y=j) = P(X=i, | N=i+j)P(N=i+j)$$






      I omitted the remaining calculations as they are irrelevant to my question. The sentence that I put in bold print is the particular sentence that is confusing me.



      How are $X$ and $Y$ - the number of eggs that hatch and don't hatch, respectively - the same events?



      My initial understanding would be that due to the strong dependent relationship they have (i.e. knowing the value of one random variable automatically gives us the value of the other), we are able to reach this conclusion.



      Is my assumption correct? Perhaps the way that the sentence was worded is confusing me.






      Any feedback is appreciated. Thank you.










      share|cite|improve this question













      I was studying probability theory using the textbook Introduction to Probability (2e) - Blitzstein & Hwang. I came across an example problem from a chapter regarding join distributions and had a question regarding a step in the solution. Here's the specific example question that I'm referring to:






      Suppoe a chicken lays a random number of eggs, $N$ where $N$ follows Pois($lambda$). Each egg independently hatches with probability $p$. Let $X$ be the number of eggs that hatch and $Y$ be the number of eggs that fail to hatch. What is the joint PMF of $X$ and $Y$?






      Textbook Solution (paraphrased):



      Because the number of eggs depends on $N$, the total number of eggs laid, we must condition on $N$.



      $$P(X=i, Y=j) = sum_{n = 0}^infty P(X=i, Y=j | N=n)P(N=n)$$



      Among the possible values of $n$, the only one that is relevant is when $n=i + j$ due to the fact that any other case would not be consistent with the number of eggs hatched and the number of eggs that failed to hatch, since $X+Y$ must equal $N$. Therefore, we can drop all of the other terms in the summation, which leaves us with,



      $$P(X=i, Y=j) = P(X=i, Y=j | N=i + j)P(N=i+j)$$



      Conditional on $N=i+j$, the events $X=i$ and $Y=j$ are exactly the same event. We can therefore drop either $X$ or $Y$ and find the joint PMF with,



      $$P(X=i, Y=j) = P(X=i, | N=i+j)P(N=i+j)$$






      I omitted the remaining calculations as they are irrelevant to my question. The sentence that I put in bold print is the particular sentence that is confusing me.



      How are $X$ and $Y$ - the number of eggs that hatch and don't hatch, respectively - the same events?



      My initial understanding would be that due to the strong dependent relationship they have (i.e. knowing the value of one random variable automatically gives us the value of the other), we are able to reach this conclusion.



      Is my assumption correct? Perhaps the way that the sentence was worded is confusing me.






      Any feedback is appreciated. Thank you.







      probability independence






      share|cite|improve this question













      share|cite|improve this question











      share|cite|improve this question




      share|cite|improve this question










      asked Nov 17 at 7:13









      Sean

      23510




      23510






















          1 Answer
          1






          active

          oldest

          votes

















          up vote
          1
          down vote



          accepted










          The given statement means that $P(X=i)=P(Y=j)$ provided that $i+j=N$ is a constant. Now from the description of $X$ and $Y$, $Xsimoperatorname{binom}(N,p)$ and $Ysimoperatorname{binom}(N,1-p)$, so
          $$P(X=i)=binom{i+j}ip^i(1-p)^j$$
          and
          $$P(Y=j)=binom{i+j}j(1-p)^jp^i$$
          It is well-known that $binom{i+j}i=binom{i+j}j$. The equality follows.






          share|cite|improve this answer





















          • Thank you so much for the clarification. I didn't think about how each random variable's PMF would look like.
            – Sean
            Nov 17 at 7:41











          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',
          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%2f3002053%2fdetermining-whether-two-events-are-the-same-or-not%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








          up vote
          1
          down vote



          accepted










          The given statement means that $P(X=i)=P(Y=j)$ provided that $i+j=N$ is a constant. Now from the description of $X$ and $Y$, $Xsimoperatorname{binom}(N,p)$ and $Ysimoperatorname{binom}(N,1-p)$, so
          $$P(X=i)=binom{i+j}ip^i(1-p)^j$$
          and
          $$P(Y=j)=binom{i+j}j(1-p)^jp^i$$
          It is well-known that $binom{i+j}i=binom{i+j}j$. The equality follows.






          share|cite|improve this answer





















          • Thank you so much for the clarification. I didn't think about how each random variable's PMF would look like.
            – Sean
            Nov 17 at 7:41















          up vote
          1
          down vote



          accepted










          The given statement means that $P(X=i)=P(Y=j)$ provided that $i+j=N$ is a constant. Now from the description of $X$ and $Y$, $Xsimoperatorname{binom}(N,p)$ and $Ysimoperatorname{binom}(N,1-p)$, so
          $$P(X=i)=binom{i+j}ip^i(1-p)^j$$
          and
          $$P(Y=j)=binom{i+j}j(1-p)^jp^i$$
          It is well-known that $binom{i+j}i=binom{i+j}j$. The equality follows.






          share|cite|improve this answer





















          • Thank you so much for the clarification. I didn't think about how each random variable's PMF would look like.
            – Sean
            Nov 17 at 7:41













          up vote
          1
          down vote



          accepted







          up vote
          1
          down vote



          accepted






          The given statement means that $P(X=i)=P(Y=j)$ provided that $i+j=N$ is a constant. Now from the description of $X$ and $Y$, $Xsimoperatorname{binom}(N,p)$ and $Ysimoperatorname{binom}(N,1-p)$, so
          $$P(X=i)=binom{i+j}ip^i(1-p)^j$$
          and
          $$P(Y=j)=binom{i+j}j(1-p)^jp^i$$
          It is well-known that $binom{i+j}i=binom{i+j}j$. The equality follows.






          share|cite|improve this answer












          The given statement means that $P(X=i)=P(Y=j)$ provided that $i+j=N$ is a constant. Now from the description of $X$ and $Y$, $Xsimoperatorname{binom}(N,p)$ and $Ysimoperatorname{binom}(N,1-p)$, so
          $$P(X=i)=binom{i+j}ip^i(1-p)^j$$
          and
          $$P(Y=j)=binom{i+j}j(1-p)^jp^i$$
          It is well-known that $binom{i+j}i=binom{i+j}j$. The equality follows.







          share|cite|improve this answer












          share|cite|improve this answer



          share|cite|improve this answer










          answered Nov 17 at 7:25









          Parcly Taxel

          41k137198




          41k137198












          • Thank you so much for the clarification. I didn't think about how each random variable's PMF would look like.
            – Sean
            Nov 17 at 7:41


















          • Thank you so much for the clarification. I didn't think about how each random variable's PMF would look like.
            – Sean
            Nov 17 at 7:41
















          Thank you so much for the clarification. I didn't think about how each random variable's PMF would look like.
          – Sean
          Nov 17 at 7:41




          Thank you so much for the clarification. I didn't think about how each random variable's PMF would look like.
          – Sean
          Nov 17 at 7:41


















          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.





          Some of your past answers have not been well-received, and you're in danger of being blocked from answering.


          Please pay close attention to the following guidance:


          • 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.


          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%2f3002053%2fdetermining-whether-two-events-are-the-same-or-not%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

          AnyDesk - Fatal Program Failure

          How to calibrate 16:9 built-in touch-screen to a 4:3 resolution?

          QoS: MAC-Priority for clients behind a repeater