Expected value and Variance in Poisson Process 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!
 A: 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 $\mathsf{Var}(Y)=\tfrac 13(3^2+4^2+5^2)-\tfrac 19(3+4+5)^2=\tfrac 23$ 
