The chemical master equation dP(n,t)/dt = λP(n−1) − (λ+μn)P(n) + μ(n+1)P(n+1) governs stochastic birth-death kinetics. The steady-state is Poissonian (λ/μ) for linear rates; nonlinear interactions broaden, skew, or create bimodal distributions unavailable to deterministic ODEs.