Kill Probability when the Weapon Bias is Randomly Distributed

This paper presents derivations of formulas for the probability of killing a point target and for the expected coverage of a population of targets when the bias of the weapon system is randomly distributed in accordance with a prescribed density function. The paper considers six cases in which the bias distributions are the gamma (2, 3), beta (2, 3), Maxwell-Boltzman (2) and Rayleigh (3), where the numbers in parentheses indicate the dimensionality of the associated kill and coverage problem. The concept of a random bias is important in situations where a unimodal distribution is perturbed by forced scattering (random or deterministic) into a density whose mode lies along a locus of points (2 dimensional) or on a surface (3 dimensional). The notion is also important when a sampling (or an a priori analysis) of the weapon system indicates that the bias is randomly distributed rather than fixed or when the concept of target evacuation is involved, in determining weapon effectiveness. The analysis in this paper is one which employs integration and special functions in the course of deriving kill and coverage formulas. Conceptually, the discussion is based upon extremely rudimentary principles of probability theory and makes no contributions to this latter field.