Article

The Moi Formula for Improper Exponential Definite Integrals

SAGE Publications Inc
Perceptual and Motor Skills
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Abstract

-A general formula is developed for solving a type of improper exponential definite integral of order n in the number plane. Termed the Moi Formula, it is shown to produce substantially simpler derivations of the finite moments of a probability distribution employed for assessing stochastic randomness, such as recently published by the authors. Other applications of the integral formula are discussed. In a recent paper a statistical method was presented for determining whether points distributed in the plane are characterized by stochastic randomness (O'Brien, 1994). O'Brien, Nguyen, and Hammel (1994) subsequently generalized the randomness method to any finite Euclidean dimension and identified areas in which the method may be employed. O'Brien (1994) demonstrated the detailed derivation of the component method measures. Those measures, derived from a spatial Poisson process, were the statistical moments used in the central limit theorem-based approximation of a normal distribution for finite samples to assess the randomness hypothesis. The purpose of the present paper is to present a recently derived formula which simplifies substantially the derivations in the above papers. The formula derived here has uses besides the specific application to O'Brien's probability method and these are discussed briefly in this paper. The formula derived in this paper is not claimed as unique for it could possibly be inferred from similar forms found in standard integration handbooks such as Gradshteyn and Ryzhik (196511980); however, it does not appear to be stated in standard handbooks for powers beyond two. Moreover, the formula derivation involves technical considerations that are beyond the scope of the present paper (such as convergence of the improper integral). Standard references such as Sokolnikoff (1939) may be consulted for discussion of these theoretical mathematical issues.

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