The inverse gamma-difference distribution and its first moment in the Cauchy principal value sense

In this paper, the probability density and distribution functions for the reciprocal-difference of independent gamma random variables with unequal shape parameters are derived. A theorem is developed and applied to evaluate the first moment of this distribution in the sense of the Cauchy principal value, which addresses the inverse chi-squared and inverse exponential-difference distributions as special cases. These results are used to find the first moment and an approximation to the centralized inverse-Fano distribution, which models the sampling distribution of the photon transfer conversion gain measurement of electro-optical imaging sensors. A Monte Carlo simulation is performed to show how the first moment of the inverse gamma-difference distribution can be utilized to control the bias of the conversion gain measurement in a live experiment. The low illumination problem of conversion gain measurement is introduced with a discussion motivating future application of the theoretical results derived.

Keywords

Cauchy principal value, centralized inverse-Fano distribution, conversion gain, first negative moment, gamma-difference distribution, generalized central limit theorem, hypergeometric function, photon transfer

2010 Mathematics Subject Classification

Primary 26A03, 30E20, 78M99. Secondary 33C20, 46N30, 62E17.

Full Text (PDF format)

Received 25 June 2018

Published 4 June 2019