Data analysis in the shot noise limit. 1. Single parameter estimation with Poisson and normal probability density functions

This paper describes some of the basic statistics required to analyze data that has random variables dlstributed according to shot, or more appropriately quantum noise statistics. This type of noise Is described by the Poisson and normal dlstrlbution with parameters related to the parent binomlal distrlbutlon. Both the Poisson and the normal dtotrlbutlon functions are analyzed In terms of the estlmation results for analysis of replicate data from a sIngle process. Analytical solutions for the parameter that best descrlbes data Obtained by replicate measurements are determined from jolnt dlstrlbutlon functions by the maximum Ilkellhood method. Parameter estimation results are different for these two dstributions. Maximum ilkelhood parameter estimation uslng the Poisson dlstrlbutlon yields results that are equivalent to the measurement mean obtalned based on a normal dlstrlbutlon with constant variance. The normal dlstributlon results in a quadratic equatbn for the single parameter that describes both the variance and the average. The maxlmun likelhood method with the joint Poisson distribution Is also used to determine the parameter that best descrlbes the mean of a random variable distributed accordlng to an Independent parameter.