Why is the sample variance equal to the average sample size for random processes?

Question

I am learning auto-correlation function for fluorescence correlation spectroscopy (FCS) on fcsXpert.com. The web page says:

For random processes such as diffusion, the average of the square of the fluctuations in a parameter equals the average parameter value.

To provide some context to this problem: in FCS, a sub-micrometer size volume in a dilute solution is illuminated with light. The solute molecules tagged with fluorescent probes diffuse across the small illuminated volume, emitting light signals upon being excited. The fluctuations to the intensity signals as the time passes are analysed using auto-correlation function. As expected, random diffusion contributes to the fluctuations to the average number of tagged molecules in the illumination volume, hence also to the fluctuations in light intensity. The question I have is:

why is the sample variance to the average molecule number in the small volume equal to the average molecule number, for random diffusion process?

Please correct me if I actually have understood the text incorrectly. Thanks!

Answer 1

This is not a property of all random processes, but it is a property of a Poisson process. The process you describe is Poisson if you assume that each excitation event has an independent and identical probability of occurring. Under this assumption, the number of events that occur in a given time period follows a Poisson distribution. This distribution has the special property that its variance is equal to its mean.