Solution
= Solution
A particle arriving at time $s\leq t$ survives until $t$ with probability $e^{-\mu(t-s)}$. Independently retaining each Poisson arrival with this time-dependent probability is <Poisson thinning>. Hence the number alive at time $t$ is Poisson with mean
$$
\lambda\int_0^te^{-\mu(t-s)}ds
=\frac\lambda\mu(1-e^{-\mu t}).
$$
Thus
$$
\boxed{N_t^{\rm alive}\sim\operatorname{Poisson}\left(\frac\lambda\mu(1-e^{-\mu t})\right).}
$$