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# Statistics and Its Interface

## Volume 12 (2019)

### Number 3

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

Pages: 467 – 478

DOI: https://dx.doi.org/10.4310/19-SII564

#### Author

#### Abstract

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.

Received 25 June 2018

Published 4 June 2019