Let's say I want to compare samples taken from Poisson distributions that have different values of lambda (rate). If the samples were from a normal distribution, I could convert each observation into a z-score based on the mean and standard deviation of the distribution that it was taken from. Then I could compare the z-scores to a common distribution.
How would I do this for samples taken from a Poisson distribution?
The specific use case is that I have a whole bunch of samples generated by many processes that have different poisson rates. The question I am asking is "Is the class of processes Poisson, or are they generated in a significantly more structured way?" I do not have enough data from any single process to answer this question, but I have more than enough data when I pool across all the processes.
Here is a more in depth explanation:
Place cells are neurons that fire action potentials preferentially when an animal is in a certain location.
I can calculate the rate that a given cell fires in a given location by dividing the number of spikes fired by the amount of time spent in that location. FR(x)
I have observations of how many spikes were actually fired by a particular cell during a particular time interval at a particular location.
If I had enough data, I could compare my distribution of spike counts at a particular location in a particular time interval to the poisson distribution.
However, this is impossible, because the time interval is different for every observation.
Thus, I would like a standardized statistic that I could translate each spike count observation into given the spike rate, time interval, and location corresponding to that observation. I could then pool these standardized statistics and compare them to a single standard distribution.