Faddy (1990) has conjectured that the variability of a pure birth process is increased, relative to the linear case, if the birth rates are convex and decreased if they are concave. We prove the conjecture by relating variability to the correlation structure of certain more informative versions of the process. A correlation inequality due to Harris (1977) is used to derive the necessary positive and negative correlation results.

Birth processes with piecewise linear birth rates are analysed, and numerical results suggest that, relative to the linear case, convex birth rates increase variability and concave birth rates decrease variability.

A stochastic death process in which particles of two types combine and are removed from the system at a rate proportional to the product of their numbers is derived and analyzed. Exact expressions for the probability distribution and moments are given and an approximating diffusion is obtained. Numerical calculations show close agreement between the exact solution and this diffusion even for small numbers of particles, and reveal the relatively low level of stochastic variation.