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Bulk arrivals. Customers arrive in buses. Buses arrive according to a Poisson process of rate $\alpha$, represented by $\sum_n\epsilon_{\Gamma_n}$. Number of customers in $k$th bus is a random variable $A_k$ with generating function $P(s)$. The process of bulk arrivals is represented by $$N=\sum_nA_n\epsilon_{\Gamma_n}$$ i.e., $N_t=\sum_nA_n[\epsilon_{\Gamma_n}(0,t)]$. Compute the Laplace functional of $N$.

I begin with writing down the form of the functional. $$\phi(\lambda)=E[\exp(-\lambda N_t)]=E\left[\exp\left(-\lambda \sum_{n=0}^{\infty}A_n[\epsilon_{\Gamma_n}(0,t)]\right)\right]$$ Update. I did find an answer but am unable to understand how they came up with the second equality step i.e. $$E(\exp(-\lambda\sum_{n=1}^\infty A_n\epsilon_{\Gamma_n})|\sum_{n=1}^{\infty}\epsilon_{\Gamma_n}=k)=[P(e^{-\lambda})]^k$$ where, $P$ is the generating function of $A_n$ for any $n$.

Is it okay to write $$A_n\overset{d}{=}A_1\Rightarrow A_n\epsilon_{\Gamma_n}\overset{d}{=}A_1\epsilon_{\Gamma_n}\Rightarrow \sum_nA_n\epsilon_{\Gamma_n}\overset{d}{=}\sum_nA_1\epsilon_{\Gamma_n}?$$ If it is, then the rest follows easily, doesn't it?

zaira
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Is it not appropiate to say $\sum_{n}A_n \epsilon_{\Gamma_n} $ and $\sum_{n}A_1 \epsilon_{\Gamma_n}$ are identically distributed. Let's refer to an easily understood practical example to see why this isn't the case.

Suppose we have a soda company that sells a four$-$pack of soda to its customers. Assume the amount of soda (in ounces) present in a single can of soda is $\mathcal{N}(12,0.1)$. What is the distribution for the amount of soda present in a four pack the company sells?

If $X_1,X_2,X_3,X_4\sim \mathcal{N}(12,0.1)$ represent the amount of soda present in the first, second, third, and fourth cans (respectively), then $$X_1+ \dots +X_4 \sim \mathcal{N}(48,0.4)$$ is the distribution of the amount of soda present in a four pack. However, $4X_1 \sim \mathcal{N}(48,0.16)$. Note the different in the variation about the mean.

In order to compute the Laplace functional of $N_t$, note we may write $$N_t=A_1 + \dots + A_{X_t}$$ where $X_t\sim \text{Poisson}(\alpha t)$ is the number of buses to arrive in $[0,t)$. Then $$\begin{eqnarray*}\phi_{N_t}(\lambda)&=&\mathbb{E}\left(e^{-\lambda N_t}\right) \\ &=& \mathbb{E}\left(\mathbb{E}\left(e^{-\lambda N_t}|X_t\right)\right) \\ &=& \mathbb{E}\left(\mathbb{E}\left(e^{-\lambda A_1}|X_t\right)\times \dots \times \mathbb{E}\left(e^{-\lambda A_{X_t}}|X_t\right)\right) \\ &=& \mathbb{E}\left(\left(P(e^{-\lambda })\right)^{X_t}\right) \\ &=& \exp\Big\{\alpha t\left(P(e^{-\lambda})-1\right)\Big\}\end{eqnarray*}$$