A blanket manufacturer is considering two new looms for his factory, loom A and loom B. The number of times that loom A has to be retuned per day is a random variable A which has a Poisson distribution with parameter $0.2t$,where t denotes the number of hours of daily operation. The number of times that loom B has to be retuned per day is a random variable B which has a Poisson distribution with parameter $0.15t$. Suppose that the daily cost of running loom A is $2t+10A^2$ and the daily cost of running loom B is $4t +10B^2$.
a)Write down E[A]
b)Write down E[A$^2$]
c)Write down E[B]
d)Write down E[B$^2$]
e)Find the the expected daily cost of loom A in terms of t.
f) Find the the expected daily cost of loom B in terms of t.
Have not met poisson distribution with parameters like this yet, happy to try the rest if someone could show and explain their result to E[A] and E[A$^2$].
a) You know that the mean of a Poisson with parameter $\lambda$ is $\lambda$. So the mean of a Poisson with parameter $0.2t$ is $0.2t$.
b) You may recall the variance of a Poisson with parameter $\lambda$: it is $\lambda$. Since in general $\text{Var}(W)=E(W^2)-(E(W))^2$, we have $E(A^2)=0.2t+(0.2t)^2$.
e) The daily cost of loom A is $2t+10A^2$. Call this random variable $X$. Then $E(X)=2t+10E(A^2)$. Now use the result of b).