= Solution
A rate-$\lambda$ <Poisson process> is a counting process starting at zero, with right-continuous nondecreasing integer-valued paths, independent stationary increments, and $N_t-N_s\sim\operatorname{Pois}(\lambda(t-s))$ for $0\leq s<t$. Its jumps have size one almost surely.
For independent <Poisson processes> $N,M$, increments of $N+M$ on disjoint intervals are independent, because all the corresponding increments of the two processes are jointly independent. On an interval of length $t$, multiplying their <probability generating functions> gives
$$
\exp(\lambda t(z-1))\exp(\mu t(z-1))=\exp((\lambda+\mu)t(z-1)).
$$
Thus the increment is Poisson with mean $(\lambda+\mu)t$. The path requirements hold, and independent processes almost surely have no simultaneous jumps. Therefore \b[$N+M$ is a <Poisson process> of rate $\lambda+\mu$].
Back to article page
