Expectation and variance of number of movie tickets

Question

Let $N\sim\mathrm{Pois}(\lambda_1)$ be the number of movies that will be released next year. Suppose that for each movie the number of tickets sold is $T\sim\mathrm{Pois}(\lambda_2)$, 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!

Answer 1

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
$$N \sim \mathrm{Pois}({\lambda}_1)\quad and\quad T\sim \mathrm{Pois}({\lambda}_2)\; independently$$ 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 T\sim \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