  Suppose that a point-like steady source at $x=0$ injects particles into a
half-infinite line. The particles diffuse and die. At long times a
non-equilibrium steady state sets in, and we assume that it involves many
particles. If the particles are non-interacting, their total number $N$ in the
steady state is Poisson-distributed with mean $\bar{N}$ predicted from a
deterministic reaction-diffusion equation. Here we determine the most likely
density history of this driven system conditional on observing a given $N$. We
also consider two prototypical examples of \emph{interacting} diffusing
particles: (i) a family of mortal diffusive lattice gases with constant
diffusivity (as illustrated by the simple symmetric exclusion process with
mortal particles), and (ii) random walkers that can annihilate in pairs. In
both examples we calculate the variances of the (non-Poissonian) stationary
distributions of $N$.
