At time $0$, cars are positioned along an infinite highway according to a homogeneous PP with rate $\alpha$. Assume the initial position of the $n$th car is $X_n$. Each car chooses a velocity independently and proceeds to travel at that fixed velocity. Say, the velocities are $\{V_n\}$ and assume that they are iid random variables with state space $(-\infty,\infty)$. Collisions are assumed impossible and cars can pass through each other. Assume $E|V_1|<\infty.$ Let $N_0(.)$ be the PP of the initial positions and $N_t(.)$ be the point process describing positions at time $t$. If $N_t(.)$ is a poisson process, find its mean meaure?
The positions at time $t$ are $X_n-tV_n$. $N_0$ is PRM with rate $\alpha$, i.e. $$N_0((a,b))=\sum_n\epsilon_{X_n}((a,b))\sim\text{Poi}(\alpha(b-a)).$$ For $N_t$, $$N_t((a,b))=\sum_n\epsilon_{X_n+tV_n}((a,b))$$ $$X_n+tV_n\overset{d}{=}X_n+tV_1\Rightarrow P(a<X_n+tV_n<b)=P(a-tV_1<X_n<b-tV_1)$$ Is this even correct? I followed this up with $$\mu_0(a,b)=EN_0(a,b)=\sum_{n=1}^\infty E(\epsilon_{X_n(a,b)})=\sum_{n=1}^{\infty}P(X_n\in(a,b))\text{ and,}$$ $$\mu_t(a,b)=EN_t(a,b)=\sum_{n=1}^\infty E(\epsilon_{X_n+tV_n(a,b)})=\sum_{n=1}^{\infty}P(X_n\in(a-tV_1,b-tV_1))=\mu_0(a-tV_1,b-tV_1)$$ Where do I go from here?