= Solution
In an interval of length $t$, condition on the <Poisson process> count $X_t=k$. Independently marking its points gives $Y_t\mid X_t=k\sim\operatorname{Bin}(k,p)$. The joint <probability generating function> of kept and discarded counts is
$$
\mathbb E[s^{Y_t}u^{X_t-Y_t}]=\mathbb E[(ps+(1-p)u)^{X_t}]=\exp\{\lambda t[p(s-1)+(1-p)(u-1)]\}.
$$
It factors into the generating functions of independent Poisson variables of means $\lambda pt$ and $\lambda(1-p)t$. The same calculation applies to any interval, and counts and marks in disjoint intervals are independent. Hence \b[$Y$ is a Poisson process of intensity $\boxed{\lambda p}$], and \b[$Y_t$ and $X_t-Y_t$ are independent]. This is the <Poisson thinning theorem>.
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