The mean of the compound Poisson process

Question

I have a compound Poisson process with uniform distribution. The Poisson arrival rate is $=\lambda$ and uniform distribution is defined over $[0,a]$. What will be the mean of this compound Poisson process with uniform distribution? Thanks in advance.

Answer 1

You may use conditional expectation.

Suppose $$ Y_t=\sum_{j=1}^{N_t}U_i, $$ where $N_t$ is a Poisson process with parameter $\lambda$, and each $U_j\sim\text{Unif}(\left[0,a\right])$. Here we suppose that $\left\{U_j\right\}_{j=1}^{\infty}$ is an independent sequence of random variables, and that it is independent from the process $N_t$. Then \begin{align} \mathbb{E}Y_t&=\mathbb{E}\left(\mathbb{E}\left(Y_t|N_t\right)\right)\\ &=\mathbb{E}\left(\mathbb{E}\left(\sum_{j=1}^{N_t}U_i\Bigg|N_t\right)\right)\\ &=\mathbb{E}\left(\sum_{j=1}^{N_t}\mathbb{E}\left(U_j|N_t\right)\right)\\ &=\mathbb{E}\left(\sum_{j=1}^{N_t}\mathbb{E}U_j\right)\\ &=\mathbb{E}\left(\sum_{j=1}^{N_t}\frac{a}{2}\right)\\ &=\frac{a}{2}\mathbb{E}N_t\\ &=\frac{a}{2}\lambda t. \end{align}