The Coefficient of Variation (CV) according to the linked Wikipedia article is the ratio of the standard deviation $\sigma$ to the mean $\mu$ of a probability distribution.
For the Poisson Distribution, the mean is equal to the variance $\sigma^2$ so that the CV should be $\sigma/\mu=\sigma/\sigma^2=1/\sigma$
However, in the calcium signaling modeling literature I am seeing some papers that state this differently. One such example is in figure 2B of this paper, although similar language is found in other papers by these researchers and their affiliates in models of the $\text{IP}_3$ receptors. Below is the figure and description, where the bold part is my own emphasis. (ISI stand for "Inter-Spike Interval", which is a stochastic time interval measured from the system in question).
B: Relationship between the standard deviation and the mean of experimental ISIs. Data obtained from 14 ASMC in 5 mouse lung slices. The relationship is approximately linear with a slope of 0.66, which implies that an inhomogeneous Poisson process governs the generation of oscillations. The dashed line indicates where the coefficient of variation (CV) is 1 (as it is for a pure Poisson process).
The paper says that the CV for a Poisson process is $1$. But this doesn't make sense given the above information. If they compared the variance to the mean then this should work out fine since their ratio is $1$, but this is done in many papers by these / affiliated researchers (that are some key people in the field), so I am not sure what is going on here.
Am I missing something with the definition of the CV or Poisson distributions? Also, how does this work with quantities with physical units where the mean and variance do not have the same dimensions? How could the mean be equal to the variance in situations like this?
Best Answer
If the spike events are modeled as a Poisson process with constant rate, then the ISIs, i.e. the time between spike events, are exponentially distributed. An exponential random variable has a mean and standard deviation equal to each other, and thus a CV of 1. Since the figure shows an estimated CV less than 1, they state this implies that their Poisson process would be better modeled with a non-constant rate (inhomogenous).
The units of mean and standard deviation for any random variable will be the same irrespective of the application.