I was looking for some help with this question I came across.
The purchase of coffee from a cafe follows a Poisson process with rate λ = 300 coffees per day. The price Y of each coffee purchased is uniformly distributed on {3, 4, 5}.
The total amount of money the cafe makes in time t is defined $$S(t) = \sum_{i=1}^{N(t)} Y_i$$
where Y1, . . . , $Y_{N(t)}$ are i.i.d according to Y .
(a) What is the expected amount of money the cafe will make in one week?
Here I got I got that $E(N(7)) = \lambda t = 300*7 = 2100 (<\infty)$, and $E(Y)=4$ (Is that right?) Then $$E(S(7)) = E(N(7))(E(Y) = 2100*4 =$8400$$
Is this the correct way to do this part? I feel like I might be using E(Y) incorrectly by setting it as 4. (Because of 4 being the middle of the uniform distribution)
(b) What is the variance of the same? I got $$E(N(7)^2) = Var(n(7)) + E(N(7))^2 = 7\lambda +7^2\lambda ^2 (<\infty)$$ Then $Var(Y) = 4^2$, $Var(N(7))=E(N(7))=2100$, so: $$Var(S(7))=E(N(7))Var(Y)+Var(N(7))(E(Y)^2=(2100*4^2)+(2100*4^2)$$ $$=67200$$
Is this correct?
(c) Would these results change for a Poisson process where λ(t) depends on the time of day as well? This one is more of a bonus question, I think the results would stay the same but I'm more focused on the other two questions for now
All help is appreciated, Thanks in advance!
You are using the Law of Total Expectation: $$\begin{align}\mathsf E(S{\small (7)}) ~&=\mathsf E(\mathsf E(S{\small (7)}\mid N{\small (7)})\\ &= \mathsf E(\mathsf E({\sum}_{y=1}^{N{\small (7)}} Y_i\mid N{\small (7)}))\\ & =\mathsf E(N{\small(7)}~\mathsf E(Y))\\ &=\mathsf E(N{\small (7)})~\mathsf E(Y) \\ &= 300\cdot 7\cdot 4\end{align}$$
And similarly , the Law of Total Variance: $$\mathsf {Var}(S{\small (7)})~{=\mathsf E(\mathsf{Var}(\sum_{i=1}^{N{\small (7)}} Y_i\mid N{\small (7)})+\mathsf{Var}(\mathsf E(\sum_{i=1}^{N{\small (7)}} Y_i\mid N{\small (7)})\\=\mathsf E(N{\small (7)}\mathsf{Var}(Y))+\mathsf{Var}(N{\small (7)}\mathsf E(Y))\\=\mathsf E(N{\small (7)})~\mathsf{Var}(Y)+\mathsf E(Y)^2~\mathsf{Var}(N{\small (7)})\\= 300\cdot 7\cdot \mathsf{Var}(Y)+4^2\cdot 300\cdot 7}$$ Which is entirely correct, since the $(Y_i)$ sequence are each iid to $Y$ and their count $N{\small (7)}$.
However: $Y$ is uniform discrete over contiguous integers $\mathsf{Var}(Y)= \dfrac{(5-3+1)^2-1}{12}=\dfrac{2}{3}$
or $\mathsf{Var}(Y)=\tfrac 13(3^2+4^2+5^2)-\tfrac 19(3+4+5)^2=\tfrac 23$