Fano factor

In statistics, the Fano factor, like the coefficient of variation, is a measure of the dispersion of a counting process. It was originally used to measure the Fano noise in ion detectors. It is named after Ugo Fano, an Italian American physicist. The Fano factor after a time is defined as where is the standard deviation and is the mean number of events of a counting process after some time . The Fano factor can be viewed as a kind of noise-to-signal ratio; it is a measure of the reliability with which the waiting time random variable can be estimated after several random events. For a Poisson counting process, the variance in the count equals the mean count, so . For a counting process , the Fano factor after a time is defined as, Sometimes, the long term limit is also termed the Fano factor, For a renewal process with holding times distributed similar to a random variable , we have that, Since we have that the right hand side is equal to the square of the coefficient of variation , the right hand side of this equation is sometimes referred to as the Fano factor as well. When considered as the dispersion of the number, the Fano factor roughly corresponds to the width of the peak of . As such, the Fano factor is often interpreted as the unpredictability of the underlying process. When the holding times are constant, then . As such, if then we interpret the renewal process as being very predictable. When the likelihood of an event occurring in any time interval is equal for all time, then the holding times must be exponentially distributed, giving a Poisson counting process, for which . In particle detectors, the Fano factor results from the energy loss in a collision not being purely statistical. The process giving rise to each individual charge carrier is not independent as the number of ways an atom may be ionized is limited by the discrete electron shells. The net result is a better energy resolution than predicted by purely statistical considerations.