We can write $\xi(t)$ as $$\xi(t)=\sum_{k=1}^{N(t)} \xi_k$$ where $\{N(t)\}_{t\in\mathbb{R}}$ is a counting Poisson process with rate $\lambda$. It is not specified in your question but I assume that $\lambda>0$ otherwise we would have $N(t)=0$ for all $t$ and thus $\xi(t)=0$ which contradicts what we are trying to prove. I am also going to assume that ...


If $\lambda$ is the parameter of an exponential distribution it's called the rate and the mean is $\frac 1{\lambda}$ which is expected time. So if the bus passes with exponential distribution having parameter $\lambda=2/\text{sec}$ then you expect a bus to pass in $\frac 12 $ second.

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