Although we are primarily concerned with Binomial probabilities in this blog, it is occasionally worth a detour to make a point.
A common bias I witness among researchers in discussing statistics is the intuition (presumption) that distributions are Gaussian (Normal) and symmetric. But many naturally-occurring distributions are not Normal, and a key reason is the influence of boundary conditions.
Even for ostensibly Real variables, unbounded behaviour is unusual. Nature is full of boundaries.
Consequently, mathematical models that incorporate boundaries can sometimes offer a fresh perspective on old problems. Gould (1996) discusses a prediction in evolutionary biology regarding the expected distribution of biomass for organisms of a range of complexity (or scale), from those composed of a single cell to those made up of trillions of cells, like humans. His argument captures an idea about evolution that places the emphasis not on the most complex or ‘highest stages’ of evolution (as conventionally taught), but rather on the plurality of blindly random evolutionary pathways. Life becomes more complex due to random variation and stable niches (‘local maxima’) rather than some external global tendency, such as a teleological advantage of complexity for survival.
Gould’s argument may be summarised in the following way. Through blind random Darwinian evolution, simple organisms may evolve into more complex ones (‘complexity’ measured as numbers of cells or organism size), but at the same time others may evolve into simpler, but perhaps equally successful ones. ‘Success’ here means reproductive survival – producing new organisms of the same scale or greater that survive to reproduce themselves.
His second premise is also non-controversial. Every organism must have at least one cell and all the first lifeforms were unicellular.
Now, run time’s arrow forwards. Assuming a constant and an equal rate of evolution, by simulation we can obtain a range of distributions like those in the Figure below.