Generalized Hit Probabilities with a Gaussian Target

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.