I think this is a question for the compound Poisson process which I am unable to solve. The question is as follows:
Batches of customers arrive at a shop at instants $S: Ω → {\Bbb{R}_+}^{\Bbb{Z}_+}$ where $S_0 = 0$. The random inter-arrival time sequence $X: Ω → {\Bbb{R}}^{\Bbb{N}}_+$ defined by $X_n = S_n − S_{n−1}$ for all $n ∈ N$ is assumed to be i.i.d. with $$P\{X_n >t\} = \exp(-\lambda t) \: \forall\: t\ge 0$$ A batch of $B_n$ customers arrive at the $n^{th}$ arrival instant, where $B : Ω → \{1,2,3,\dots,M\}^{\Bbb{N}}$ is an i.i.d. sequence independent of the arrival sequence, with distribution $$p_j :=P\{B_n=j\}, \:\: j\in \{1,2,3,\dots,M\}$$ Fix a $k ∈ \{1,2,3,\dots,M\}$ to define a random process $Z_k : Ω → \Bbb{Z}_+ ^{\Bbb{R}_+}$ where $Z_k(t)$ denotes the number of batches of size $k$ that arrive in the time interval $[0,t]$.
(a) Find mean of $Z_k(t)$
(b) Find $\Bbb{E}[Z_k(t)Z_l(t)]$ for $l \neq k$ and $l \in \{1,2,3,\dots,M\}$
What I understand is I need to calculate $\Bbb{E}[N(t) \mid Batch\:size = k]$ (where $N(t)$ is the associated counting process) but I don't know how to formulate it.
Any help will be appreciated.
Thanks in advance