Expectation and variance of number of movie tickets
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
edited Dec 4 '18 at 13:05
Davide Giraudo
125k16150259
asked Dec 1 '18 at 14:41
user587126
155
add a comment |
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
edited Dec 4 '18 at 13:05
Davide Giraudo
125k16150259
asked Dec 1 '18 at 14:41
user587126
155
To compute $text{Var}[NT]$, use the fact that $text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$.
– Mike Earnest
Dec 2 '18 at 22:36
But how do you know that $N^2$ and $T^2$ are independent?
– user587126
Dec 3 '18 at 13:09
Because functions of independent random variables are independent.
– Mike Earnest
Dec 3 '18 at 15:08
add a comment |
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
edited Dec 4 '18 at 13:05
Davide Giraudo
125k16150259
asked Dec 1 '18 at 14:41
user587126
155
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
edited Dec 4 '18 at 13:05
Davide Giraudo
125k16150259
asked Dec 1 '18 at 14:41
user587126
155
edited Dec 4 '18 at 13:05
Davide Giraudo
125k16150259
asked Dec 1 '18 at 14:41
user587126
155
edited Dec 4 '18 at 13:05
Davide Giraudo
125k16150259
edited Dec 4 '18 at 13:05
Davide Giraudo
125k16150259
edited Dec 4 '18 at 13:05
Davide Giraudo
125k16150259
125k16150259
asked Dec 1 '18 at 14:41
user587126
155
asked Dec 1 '18 at 14:41
user587126
155
asked Dec 1 '18 at 14:41
user587126
155
155
To compute $text{Var}[NT]$, use the fact that $text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$.
– Mike Earnest
Dec 2 '18 at 22:36
But how do you know that $N^2$ and $T^2$ are independent?
– user587126
Dec 3 '18 at 13:09
Because functions of independent random variables are independent.
– Mike Earnest
Dec 3 '18 at 15:08
add a comment |
To compute $text{Var}[NT]$, use the fact that $text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$.
– Mike Earnest
Dec 2 '18 at 22:36
But how do you know that $N^2$ and $T^2$ are independent?
– user587126
Dec 3 '18 at 13:09
Because functions of independent random variables are independent.
– Mike Earnest
Dec 3 '18 at 15:08
To compute $text{Var}[NT]$, use the fact that $text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$.
– Mike Earnest
Dec 2 '18 at 22:36
To compute $text{Var}[NT]$, use the fact that $text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$.
– Mike Earnest
Dec 2 '18 at 22:36
But how do you know that $N^2$ and $T^2$ are independent?
– user587126
Dec 3 '18 at 13:09
But how do you know that $N^2$ and $T^2$ are independent?
– user587126
Dec 3 '18 at 13:09
Because functions of independent random variables are independent.
– Mike Earnest
Dec 3 '18 at 15:08
Because functions of independent random variables are independent.
– Mike Earnest
Dec 3 '18 at 15:08
add a comment |
1 Answer
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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
edited Dec 4 '18 at 13:23
Davide Giraudo
125k16150259
answered Dec 3 '18 at 14:07
Amelia
329
add a comment |
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1 Answer
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active
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1 Answer
1
active
oldest
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oldest
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active
oldest
votes
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
edited Dec 4 '18 at 13:23
Davide Giraudo
125k16150259
answered Dec 3 '18 at 14:07
Amelia
329
add a comment |
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
edited Dec 4 '18 at 13:23
Davide Giraudo
125k16150259
answered Dec 3 '18 at 14:07
Amelia
329
add a comment |
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
edited Dec 4 '18 at 13:23
Davide Giraudo
125k16150259
answered Dec 3 '18 at 14:07
Amelia
329
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
edited Dec 4 '18 at 13:23
Davide Giraudo
125k16150259
answered Dec 3 '18 at 14:07
Amelia
329
edited Dec 4 '18 at 13:23
Davide Giraudo
125k16150259
edited Dec 4 '18 at 13:23
Davide Giraudo
125k16150259
edited Dec 4 '18 at 13:23
Davide Giraudo
125k16150259
125k16150259
answered Dec 3 '18 at 14:07
Amelia
329
answered Dec 3 '18 at 14:07
Amelia
329
answered Dec 3 '18 at 14:07
Amelia
329
329
add a comment |
add a comment |
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To compute $text{Var}[NT]$, use the fact that $text{Var}[NT]=E[(NT)^2]-(E[NT])^2=E[N^2]E[T^2]-(E[N]E[T])^2$.
– Mike Earnest
Dec 2 '18 at 22:36
But how do you know that $N^2$ and $T^2$ are independent?
– user587126
Dec 3 '18 at 13:09
Because functions of independent random variables are independent.
– Mike Earnest
Dec 3 '18 at 15:08