Expectation and variance of number of movie tickets












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$begingroup$


Let $Nsimmathrm{Pois}(lambda_1)$ be the number of movies that will be released next year. Suppose that for each movie the number of tickets sold is $Tsimmathrm{Pois}(lambda_2)$, independently. Find the mean and variance of the number of movie tickets that will be sold next year.
The expectation is quite easy to find. $E(NT)=E(E(N|T))=lambda_1×lambda_2$. But I am stuck at calculating variance. Any help would be appreciated!






probability-theory conditional-expectation poisson-distribution variance







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$endgroup$












  • $begingroup$
    To compute $text{Var}[NT]$, use the fact that $text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$.
    $endgroup$
    – Mike Earnest
    Dec 2 '18 at 22:36










  • $begingroup$
    But how do you know that $N^2$ and $T^2$ are independent?
    $endgroup$
    – user587126
    Dec 3 '18 at 13:09










  • $begingroup$
    Because functions of independent random variables are independent.
    $endgroup$
    – Mike Earnest
    Dec 3 '18 at 15:08
















0












$begingroup$


Let $Nsimmathrm{Pois}(lambda_1)$ be the number of movies that will be released next year. Suppose that for each movie the number of tickets sold is $Tsimmathrm{Pois}(lambda_2)$, independently. Find the mean and variance of the number of movie tickets that will be sold next year.
The expectation is quite easy to find. $E(NT)=E(E(N|T))=lambda_1×lambda_2$. But I am stuck at calculating variance. Any help would be appreciated!






probability-theory conditional-expectation poisson-distribution variance







share|cite|improve this question











$endgroup$












  • $begingroup$
    To compute $text{Var}[NT]$, use the fact that $text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$.
    $endgroup$
    – Mike Earnest
    Dec 2 '18 at 22:36










  • $begingroup$
    But how do you know that $N^2$ and $T^2$ are independent?
    $endgroup$
    – user587126
    Dec 3 '18 at 13:09










  • $begingroup$
    Because functions of independent random variables are independent.
    $endgroup$
    – Mike Earnest
    Dec 3 '18 at 15:08














0












0








0


0



$begingroup$


Let $Nsimmathrm{Pois}(lambda_1)$ be the number of movies that will be released next year. Suppose that for each movie the number of tickets sold is $Tsimmathrm{Pois}(lambda_2)$, independently. Find the mean and variance of the number of movie tickets that will be sold next year.
The expectation is quite easy to find. $E(NT)=E(E(N|T))=lambda_1×lambda_2$. But I am stuck at calculating variance. Any help would be appreciated!






probability-theory conditional-expectation poisson-distribution variance







share|cite|improve this question











$endgroup$




Let $Nsimmathrm{Pois}(lambda_1)$ be the number of movies that will be released next year. Suppose that for each movie the number of tickets sold is $Tsimmathrm{Pois}(lambda_2)$, independently. Find the mean and variance of the number of movie tickets that will be sold next year.
The expectation is quite easy to find. $E(NT)=E(E(N|T))=lambda_1×lambda_2$. But I am stuck at calculating variance. Any help would be appreciated!






probability-theory conditional-expectation poisson-distribution variance




probability-theory conditional-expectation poisson-distribution variance






share|cite|improve this question















share|cite|improve this question













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share|cite|improve this question








edited Dec 4 '18 at 13:05









Davide Giraudo

126k16151263




126k16151263










asked Dec 1 '18 at 14:41









user587126user587126

155




155












  • $begingroup$
    To compute $text{Var}[NT]$, use the fact that $text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$.
    $endgroup$
    – Mike Earnest
    Dec 2 '18 at 22:36










  • $begingroup$
    But how do you know that $N^2$ and $T^2$ are independent?
    $endgroup$
    – user587126
    Dec 3 '18 at 13:09










  • $begingroup$
    Because functions of independent random variables are independent.
    $endgroup$
    – Mike Earnest
    Dec 3 '18 at 15:08


















  • $begingroup$
    To compute $text{Var}[NT]$, use the fact that $text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$.
    $endgroup$
    – Mike Earnest
    Dec 2 '18 at 22:36










  • $begingroup$
    But how do you know that $N^2$ and $T^2$ are independent?
    $endgroup$
    – user587126
    Dec 3 '18 at 13:09










  • $begingroup$
    Because functions of independent random variables are independent.
    $endgroup$
    – Mike Earnest
    Dec 3 '18 at 15:08
















$begingroup$
To compute $text{Var}[NT]$, use the fact that $text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$.
$endgroup$
– Mike Earnest
Dec 2 '18 at 22:36




$begingroup$
To compute $text{Var}[NT]$, use the fact that $text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$.
$endgroup$
– Mike Earnest
Dec 2 '18 at 22:36












$begingroup$
But how do you know that $N^2$ and $T^2$ are independent?
$endgroup$
– user587126
Dec 3 '18 at 13:09




$begingroup$
But how do you know that $N^2$ and $T^2$ are independent?
$endgroup$
– user587126
Dec 3 '18 at 13:09












$begingroup$
Because functions of independent random variables are independent.
$endgroup$
– Mike Earnest
Dec 3 '18 at 15:08




$begingroup$
Because functions of independent random variables are independent.
$endgroup$
– Mike Earnest
Dec 3 '18 at 15:08










1 Answer
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$begingroup$

Given,
$N$=Number of movies that will be released next year


and $T$=number of tickets sold for each movie


we are also given that

$$N sim mathrm{Pois}({lambda}_1)quad andquad Tsim mathrm{Pois}({lambda}_2); independently$$
We now need to find out the mean and variance of the number of movie tickets that will be sold next year.


Number of movie tickets that will be sold next year=NT

Thus we have to find E[NT] and Var[NT]

(since N and T are independent Random variables)
$$E[NT]=E[N]E[T]={lambda_1 lambda_2}$$

(since $N sim mathrm{Pois}({lambda}_1)mbox{ and }quad Tsim mathrm{Pois}({lambda}_2);;E[N]=lambda_1;,E[T]= lambda_2) $

Now;



$$text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$$
[since $N$ and $T$ are independent $Random ;variables$ then $g(N)$ and $f(T)$ are independent.
Thus $N^2$ and $T^2$ are independent Random variables]


Refer:Are functions of independent variables also independent?


Now just evaluate$E[N^2];and;E[T^2]$.Then you are done






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    1












    $begingroup$

    Given,
    $N$=Number of movies that will be released next year


    and $T$=number of tickets sold for each movie


    we are also given that

    $$N sim mathrm{Pois}({lambda}_1)quad andquad Tsim mathrm{Pois}({lambda}_2); independently$$
    We now need to find out the mean and variance of the number of movie tickets that will be sold next year.


    Number of movie tickets that will be sold next year=NT

    Thus we have to find E[NT] and Var[NT]

    (since N and T are independent Random variables)
    $$E[NT]=E[N]E[T]={lambda_1 lambda_2}$$

    (since $N sim mathrm{Pois}({lambda}_1)mbox{ and }quad Tsim mathrm{Pois}({lambda}_2);;E[N]=lambda_1;,E[T]= lambda_2) $

    Now;



    $$text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$$
    [since $N$ and $T$ are independent $Random ;variables$ then $g(N)$ and $f(T)$ are independent.
    Thus $N^2$ and $T^2$ are independent Random variables]


    Refer:Are functions of independent variables also independent?


    Now just evaluate$E[N^2];and;E[T^2]$.Then you are done






    share|cite|improve this answer











    $endgroup$


















      1












      $begingroup$

      Given,
      $N$=Number of movies that will be released next year


      and $T$=number of tickets sold for each movie


      we are also given that

      $$N sim mathrm{Pois}({lambda}_1)quad andquad Tsim mathrm{Pois}({lambda}_2); independently$$
      We now need to find out the mean and variance of the number of movie tickets that will be sold next year.


      Number of movie tickets that will be sold next year=NT

      Thus we have to find E[NT] and Var[NT]

      (since N and T are independent Random variables)
      $$E[NT]=E[N]E[T]={lambda_1 lambda_2}$$

      (since $N sim mathrm{Pois}({lambda}_1)mbox{ and }quad Tsim mathrm{Pois}({lambda}_2);;E[N]=lambda_1;,E[T]= lambda_2) $

      Now;



      $$text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$$
      [since $N$ and $T$ are independent $Random ;variables$ then $g(N)$ and $f(T)$ are independent.
      Thus $N^2$ and $T^2$ are independent Random variables]


      Refer:Are functions of independent variables also independent?


      Now just evaluate$E[N^2];and;E[T^2]$.Then you are done






      share|cite|improve this answer











      $endgroup$
















        1












        1








        1





        $begingroup$

        Given,
        $N$=Number of movies that will be released next year


        and $T$=number of tickets sold for each movie


        we are also given that

        $$N sim mathrm{Pois}({lambda}_1)quad andquad Tsim mathrm{Pois}({lambda}_2); independently$$
        We now need to find out the mean and variance of the number of movie tickets that will be sold next year.


        Number of movie tickets that will be sold next year=NT

        Thus we have to find E[NT] and Var[NT]

        (since N and T are independent Random variables)
        $$E[NT]=E[N]E[T]={lambda_1 lambda_2}$$

        (since $N sim mathrm{Pois}({lambda}_1)mbox{ and }quad Tsim mathrm{Pois}({lambda}_2);;E[N]=lambda_1;,E[T]= lambda_2) $

        Now;



        $$text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$$
        [since $N$ and $T$ are independent $Random ;variables$ then $g(N)$ and $f(T)$ are independent.
        Thus $N^2$ and $T^2$ are independent Random variables]


        Refer:Are functions of independent variables also independent?


        Now just evaluate$E[N^2];and;E[T^2]$.Then you are done






        share|cite|improve this answer











        $endgroup$



        Given,
        $N$=Number of movies that will be released next year


        and $T$=number of tickets sold for each movie


        we are also given that

        $$N sim mathrm{Pois}({lambda}_1)quad andquad Tsim mathrm{Pois}({lambda}_2); independently$$
        We now need to find out the mean and variance of the number of movie tickets that will be sold next year.


        Number of movie tickets that will be sold next year=NT

        Thus we have to find E[NT] and Var[NT]

        (since N and T are independent Random variables)
        $$E[NT]=E[N]E[T]={lambda_1 lambda_2}$$

        (since $N sim mathrm{Pois}({lambda}_1)mbox{ and }quad Tsim mathrm{Pois}({lambda}_2);;E[N]=lambda_1;,E[T]= lambda_2) $

        Now;



        $$text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$$
        [since $N$ and $T$ are independent $Random ;variables$ then $g(N)$ and $f(T)$ are independent.
        Thus $N^2$ and $T^2$ are independent Random variables]


        Refer:Are functions of independent variables also independent?


        Now just evaluate$E[N^2];and;E[T^2]$.Then you are done







        share|cite|improve this answer














        share|cite|improve this answer



        share|cite|improve this answer








        edited Dec 4 '18 at 13:23









        Davide Giraudo

        126k16151263




        126k16151263










        answered Dec 3 '18 at 14:07









        AmeliaAmelia

        329




        329






























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