Exact poisson distribution in the use of NPB with aiming errors

A neutral particle beam(NPB) aimed at the object and receive a small number of neutron signals at the detector to estimate the mass of an object Since there is uncertainty about the location of the axis of the beam relative to the object, we could have aiming errors which may lead to incorrect information about the object. Under the two assumptions that neutral particle scattering distribution and aiming errors have a circular normal distribution respectively, we have derived an exact probability distribution of neutral particles. It becomes a Poison-power function distribution., We proved monotone likelihood ratio property of tlis distribution. This property can be used to find a criteria for the hypothesis testing problem.