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June, 1951 Generalized Hit Probabilities with a Gaussian Target
D. A. S. Fraser
Ann. Math. Statist. 22(2): 248-255 (June, 1951). DOI: 10.1214/aoms/1177729644


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.


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D. A. S. Fraser. "Generalized Hit Probabilities with a Gaussian Target." Ann. Math. Statist. 22 (2) 248 - 255, June, 1951.


Published: June, 1951
First available in Project Euclid: 28 April 2007

zbMATH: 0043.13302
MathSciNet: MR40620
Digital Object Identifier: 10.1214/aoms/1177729644

Rights: Copyright © 1951 Institute of Mathematical Statistics

Vol.22 • No. 2 • June, 1951
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