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Mean and Variance of Continuous Random Variable


Mean and variance of call center dataPoisson random variable- varianceHow to obtain variance of a random variable that depends on a hypergeometric variable?Variance reduction technique in Monte Carlo integrationVariance of a continuous uniformly distributed random variableWeighted sample mean and variance - asking for references and detailsCan the variance of a continuous random variable with known distribution be impossible to find?Mean and variance of the maximum of a random number of Uniform variablesApproximating the expected value and variance of the function of a (continuous univariate) random variableMean of maximum of exponential random variables (independent but not identical)






.everyoneloves__top-leaderboard:empty,.everyoneloves__mid-leaderboard:empty,.everyoneloves__bot-mid-leaderboard:empty margin-bottom:0;








2












$begingroup$


I have a problem on my homework about the continuous random variable $y$ where the cdf is $F(y)=frac1(1+e^-y)$.



Part a is asking for the pdf which I found to be $frace^y(e^y+1)^2$.



Part b asks for the mean and variance of $y$ but when I tried to find the $E(y)$, I got zero with the integral from $-infty$ to $infty$ of $fracye^y(e^y+1)^2$. I'm not sure where I'm going wrong with this problem?










share|cite|improve this question









New contributor




EBusch is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
Check out our Code of Conduct.







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    2












    $begingroup$


    I have a problem on my homework about the continuous random variable $y$ where the cdf is $F(y)=frac1(1+e^-y)$.



    Part a is asking for the pdf which I found to be $frace^y(e^y+1)^2$.



    Part b asks for the mean and variance of $y$ but when I tried to find the $E(y)$, I got zero with the integral from $-infty$ to $infty$ of $fracye^y(e^y+1)^2$. I'm not sure where I'm going wrong with this problem?










    share|cite|improve this question









    New contributor




    EBusch is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
    Check out our Code of Conduct.







    $endgroup$














      2












      2








      2





      $begingroup$


      I have a problem on my homework about the continuous random variable $y$ where the cdf is $F(y)=frac1(1+e^-y)$.



      Part a is asking for the pdf which I found to be $frace^y(e^y+1)^2$.



      Part b asks for the mean and variance of $y$ but when I tried to find the $E(y)$, I got zero with the integral from $-infty$ to $infty$ of $fracye^y(e^y+1)^2$. I'm not sure where I'm going wrong with this problem?










      share|cite|improve this question









      New contributor




      EBusch is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.







      $endgroup$




      I have a problem on my homework about the continuous random variable $y$ where the cdf is $F(y)=frac1(1+e^-y)$.



      Part a is asking for the pdf which I found to be $frace^y(e^y+1)^2$.



      Part b asks for the mean and variance of $y$ but when I tried to find the $E(y)$, I got zero with the integral from $-infty$ to $infty$ of $fracye^y(e^y+1)^2$. I'm not sure where I'm going wrong with this problem?







      variance mean






      share|cite|improve this question









      New contributor




      EBusch is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.











      share|cite|improve this question









      New contributor




      EBusch is a new contributor to this site. Take care in asking for clarification, commenting, and answering.
      Check out our Code of Conduct.









      share|cite|improve this question




      share|cite|improve this question








      edited 3 hours ago









      Noah

      3,6011417




      3,6011417






      New contributor




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      asked 5 hours ago









      EBuschEBusch

      112




      112




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      New contributor





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          2 Answers
          2






          active

          oldest

          votes


















          1












          $begingroup$

          What makes you think you did something wrong?



          beginalign
          & Pr(Yle y) = F(y) = frac 1 1+e^-y \[10pt]
          textand & Pr(Yge -y) = 1-F(-y) = 1- frac 1 1+e^y \[8pt]
          = & frace^y1+e^y = frace^ycdot e^-y(1+e^y)cdot e^-y = frac 1 e^-y+1,
          endalign

          and therefore
          $$
          Pr(Yle y) = Pr(Y ge -y).
          $$

          So this distribution is symmetric about $0.$



          Therefore, if the expected value exists, it is $0.$



          You can also show that the density function is an even function:
          beginalign
          f(y) & = frace^y(1+e^y)^2. \[12pt]
          f(-y) & = frace^-y(1+e^-y)^2 = frace^-ycdotleft( e^y right)^2Big((1+e^-y) cdot e^y Big)^2 = frace^y(e^y+1)^2 = f(y).
          endalign

          Since the density is an even function, the expected value must be $0$ if it exists.



          The expected value $operatorname E(Y)$ exists if $operatorname E(|Y|) < +infty.$






          share|cite|improve this answer









          $endgroup$




















            0












            $begingroup$

            Comment:



            Setting what I take to be your CDF equal to $U sim mathsfUnif(0,1),$ and solving for the quantile function (inverse CDF) in terms of $U,$ I simulate a sample of a million
            observations as shown below.



            Then, when I plot one possible interpretation of your PDF through the histogram of the large sample, that density function
            seems to fit pretty well.



            set.seed(1019) # for reproducibility
            u = runif(10^6); x = -log(1/u - 1)

            hist(x, prob=T, br=100, col="skyblue2")
            curve(exp(x)/(exp(x)+1)^2, -10, 10, add=T, lwd=2, col="red")


            enter image description here



            I don't pretend that this is a 'worked answer' to your problem, but
            I hope it may give you enough clues to improve the version of the problem you posted and to finish the problem on your own.






            share|cite|improve this answer











            $endgroup$













              Your Answer





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              2 Answers
              2






              active

              oldest

              votes








              2 Answers
              2






              active

              oldest

              votes









              active

              oldest

              votes






              active

              oldest

              votes









              1












              $begingroup$

              What makes you think you did something wrong?



              beginalign
              & Pr(Yle y) = F(y) = frac 1 1+e^-y \[10pt]
              textand & Pr(Yge -y) = 1-F(-y) = 1- frac 1 1+e^y \[8pt]
              = & frace^y1+e^y = frace^ycdot e^-y(1+e^y)cdot e^-y = frac 1 e^-y+1,
              endalign

              and therefore
              $$
              Pr(Yle y) = Pr(Y ge -y).
              $$

              So this distribution is symmetric about $0.$



              Therefore, if the expected value exists, it is $0.$



              You can also show that the density function is an even function:
              beginalign
              f(y) & = frace^y(1+e^y)^2. \[12pt]
              f(-y) & = frace^-y(1+e^-y)^2 = frace^-ycdotleft( e^y right)^2Big((1+e^-y) cdot e^y Big)^2 = frace^y(e^y+1)^2 = f(y).
              endalign

              Since the density is an even function, the expected value must be $0$ if it exists.



              The expected value $operatorname E(Y)$ exists if $operatorname E(|Y|) < +infty.$






              share|cite|improve this answer









              $endgroup$

















                1












                $begingroup$

                What makes you think you did something wrong?



                beginalign
                & Pr(Yle y) = F(y) = frac 1 1+e^-y \[10pt]
                textand & Pr(Yge -y) = 1-F(-y) = 1- frac 1 1+e^y \[8pt]
                = & frace^y1+e^y = frace^ycdot e^-y(1+e^y)cdot e^-y = frac 1 e^-y+1,
                endalign

                and therefore
                $$
                Pr(Yle y) = Pr(Y ge -y).
                $$

                So this distribution is symmetric about $0.$



                Therefore, if the expected value exists, it is $0.$



                You can also show that the density function is an even function:
                beginalign
                f(y) & = frace^y(1+e^y)^2. \[12pt]
                f(-y) & = frace^-y(1+e^-y)^2 = frace^-ycdotleft( e^y right)^2Big((1+e^-y) cdot e^y Big)^2 = frace^y(e^y+1)^2 = f(y).
                endalign

                Since the density is an even function, the expected value must be $0$ if it exists.



                The expected value $operatorname E(Y)$ exists if $operatorname E(|Y|) < +infty.$






                share|cite|improve this answer









                $endgroup$















                  1












                  1








                  1





                  $begingroup$

                  What makes you think you did something wrong?



                  beginalign
                  & Pr(Yle y) = F(y) = frac 1 1+e^-y \[10pt]
                  textand & Pr(Yge -y) = 1-F(-y) = 1- frac 1 1+e^y \[8pt]
                  = & frace^y1+e^y = frace^ycdot e^-y(1+e^y)cdot e^-y = frac 1 e^-y+1,
                  endalign

                  and therefore
                  $$
                  Pr(Yle y) = Pr(Y ge -y).
                  $$

                  So this distribution is symmetric about $0.$



                  Therefore, if the expected value exists, it is $0.$



                  You can also show that the density function is an even function:
                  beginalign
                  f(y) & = frace^y(1+e^y)^2. \[12pt]
                  f(-y) & = frace^-y(1+e^-y)^2 = frace^-ycdotleft( e^y right)^2Big((1+e^-y) cdot e^y Big)^2 = frace^y(e^y+1)^2 = f(y).
                  endalign

                  Since the density is an even function, the expected value must be $0$ if it exists.



                  The expected value $operatorname E(Y)$ exists if $operatorname E(|Y|) < +infty.$






                  share|cite|improve this answer









                  $endgroup$



                  What makes you think you did something wrong?



                  beginalign
                  & Pr(Yle y) = F(y) = frac 1 1+e^-y \[10pt]
                  textand & Pr(Yge -y) = 1-F(-y) = 1- frac 1 1+e^y \[8pt]
                  = & frace^y1+e^y = frace^ycdot e^-y(1+e^y)cdot e^-y = frac 1 e^-y+1,
                  endalign

                  and therefore
                  $$
                  Pr(Yle y) = Pr(Y ge -y).
                  $$

                  So this distribution is symmetric about $0.$



                  Therefore, if the expected value exists, it is $0.$



                  You can also show that the density function is an even function:
                  beginalign
                  f(y) & = frace^y(1+e^y)^2. \[12pt]
                  f(-y) & = frace^-y(1+e^-y)^2 = frace^-ycdotleft( e^y right)^2Big((1+e^-y) cdot e^y Big)^2 = frace^y(e^y+1)^2 = f(y).
                  endalign

                  Since the density is an even function, the expected value must be $0$ if it exists.



                  The expected value $operatorname E(Y)$ exists if $operatorname E(|Y|) < +infty.$







                  share|cite|improve this answer












                  share|cite|improve this answer



                  share|cite|improve this answer










                  answered 2 hours ago









                  Michael HardyMichael Hardy

                  3,9951430




                  3,9951430























                      0












                      $begingroup$

                      Comment:



                      Setting what I take to be your CDF equal to $U sim mathsfUnif(0,1),$ and solving for the quantile function (inverse CDF) in terms of $U,$ I simulate a sample of a million
                      observations as shown below.



                      Then, when I plot one possible interpretation of your PDF through the histogram of the large sample, that density function
                      seems to fit pretty well.



                      set.seed(1019) # for reproducibility
                      u = runif(10^6); x = -log(1/u - 1)

                      hist(x, prob=T, br=100, col="skyblue2")
                      curve(exp(x)/(exp(x)+1)^2, -10, 10, add=T, lwd=2, col="red")


                      enter image description here



                      I don't pretend that this is a 'worked answer' to your problem, but
                      I hope it may give you enough clues to improve the version of the problem you posted and to finish the problem on your own.






                      share|cite|improve this answer











                      $endgroup$

















                        0












                        $begingroup$

                        Comment:



                        Setting what I take to be your CDF equal to $U sim mathsfUnif(0,1),$ and solving for the quantile function (inverse CDF) in terms of $U,$ I simulate a sample of a million
                        observations as shown below.



                        Then, when I plot one possible interpretation of your PDF through the histogram of the large sample, that density function
                        seems to fit pretty well.



                        set.seed(1019) # for reproducibility
                        u = runif(10^6); x = -log(1/u - 1)

                        hist(x, prob=T, br=100, col="skyblue2")
                        curve(exp(x)/(exp(x)+1)^2, -10, 10, add=T, lwd=2, col="red")


                        enter image description here



                        I don't pretend that this is a 'worked answer' to your problem, but
                        I hope it may give you enough clues to improve the version of the problem you posted and to finish the problem on your own.






                        share|cite|improve this answer











                        $endgroup$















                          0












                          0








                          0





                          $begingroup$

                          Comment:



                          Setting what I take to be your CDF equal to $U sim mathsfUnif(0,1),$ and solving for the quantile function (inverse CDF) in terms of $U,$ I simulate a sample of a million
                          observations as shown below.



                          Then, when I plot one possible interpretation of your PDF through the histogram of the large sample, that density function
                          seems to fit pretty well.



                          set.seed(1019) # for reproducibility
                          u = runif(10^6); x = -log(1/u - 1)

                          hist(x, prob=T, br=100, col="skyblue2")
                          curve(exp(x)/(exp(x)+1)^2, -10, 10, add=T, lwd=2, col="red")


                          enter image description here



                          I don't pretend that this is a 'worked answer' to your problem, but
                          I hope it may give you enough clues to improve the version of the problem you posted and to finish the problem on your own.






                          share|cite|improve this answer











                          $endgroup$



                          Comment:



                          Setting what I take to be your CDF equal to $U sim mathsfUnif(0,1),$ and solving for the quantile function (inverse CDF) in terms of $U,$ I simulate a sample of a million
                          observations as shown below.



                          Then, when I plot one possible interpretation of your PDF through the histogram of the large sample, that density function
                          seems to fit pretty well.



                          set.seed(1019) # for reproducibility
                          u = runif(10^6); x = -log(1/u - 1)

                          hist(x, prob=T, br=100, col="skyblue2")
                          curve(exp(x)/(exp(x)+1)^2, -10, 10, add=T, lwd=2, col="red")


                          enter image description here



                          I don't pretend that this is a 'worked answer' to your problem, but
                          I hope it may give you enough clues to improve the version of the problem you posted and to finish the problem on your own.







                          share|cite|improve this answer














                          share|cite|improve this answer



                          share|cite|improve this answer








                          edited 31 mins ago

























                          answered 4 hours ago









                          BruceETBruceET

                          6,3531721




                          6,3531721




















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