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The authors were prompted by a general problem concerning hit probabilities arising in military operations to seek the distribution of Qᵢ = ᵏ₈=₁aᵢx²ᵢ, k = 2, 3, where the xᵢ are normally and independently distributed with zero mean and unit variance, aᵢ = 1, and aᵢ > 0. While the distribution of a positive definite quadratic form in independent normal variates has been the subject of several papers in recent years 6, 11, 12, laborious computations are required to prepare from existing results the percentiles of the distribution and a table of hit probabilities. This paper discusses the exact distribution of Qₖ and then obtains and tabulates the distributions of Q₂ and Q₃, accurate to four places. Three other approaches to the distributions are discussed and compared with the exact results: a derivation by Hotelling 8, the Cornish-Fisher asymptotic approximation 3, and the approximation obtained by replacing the quadratic form with a chi-square variate whose first two moments are equated to those of the quadratic form--a type of approximation used in components of variance analysis. The exact values and the approximations are given in Tables I and II. The tables have been prepared with the original problem in mind, but also serve as an aid in several problems arising out of quite different contexts, 1, 2, 13. These are discussed in Section 6.
Grad et al. (Thu,) studied this question.