Generalized Hit Probabilities with a Gaussian Target

Abstract

A general discrete distribution is obtained whose random variable is the number of "hits" on a target. The target is $k$-dimensional and Gaussian diffuse, that is, the probability of a hit is given to within a constant factor by a Gaussian probability density function of the position of the "trajectory" in $k$ dimensions. For a series of $n$ rounds, the $n$ positions of the trajectory have a multivariate Gaussian distribution. An expression is given, using Theorems 1 and 2 or 1 and 3, for the probability of $r$ hits as a linear combination of probabilities of all hits on each possible set of rounds. Theorems 4, 5, and 6, with Theorem 1, give three limiting distributions as $n$, the number of rounds, tends to infinity. Theorems 7, 8, and 9, with Theorem 1, present three other limiting cases, and Theorems 10 and 1 give a time average result.