On Variability in Birth Processes

Let X（t）:t≥ 0 with X（0） = 0 be a pure birth process with birth rates λ_n,n ≥ 0. In this note, we prove a conjecture made by Faddy7 that VarX（t） ≥ EX（t） holds for any increasing sequence of transition rates λ₀≤λ₁≤λ₂≤ …and the reverse for any decreasing sequence.