Master Equation — Birth-Death Process

Exact probability distribution P(n,t) evolving under the birth-death master equation, compared to stochastic trajectories.

Birth rate λ 3.0 Death rate μ 1.0 Initial n₀ 0 Max state N 25

⟨n⟩ = — | Var = —

dPₙ/dt = λPₙ₋₁ + μ(n+1)Pₙ₊₁ - (λ+μn)Pₙ

Stationary: Poisson(λ/μ). Blue bars: exact P(n,t); yellow line: stochastic sample path. Mean: ⟨n(t)⟩ = n₀e^{-(μ-λ)t} + λ/μ·(1-e^{-(μ-λ)t}) for μ>λ.