The Weighted Mean Standard Deviation Distribution: A Geometrical Framework

Author(s)
R. Caimmi

ABSTRACT

The current attempt is aimed to extend previous results, concerning the explicit expression of the arithmetic mean standard deviation distribution, to the general case of the weighted mean standard deviation distribution. To this respect, the integration domain is expressed in canonical form after a change of reference frame in the n-space, which is recognized as an infinitely thin n-cylindrical corona where the axis coincides with a coordinate axis and the orthogonal section is an infinitely thin, homotetic (n-1)-elliptical corona. The semiaxes are formulated in two different ways, namely in terms of (1) eigenvalues, via the eigenvalue equation, and (2) leading principal minors of the matrix of a quadratic form, via the Jacobi formulae. The distribution and related parameters have the same formal expression with respect to their counterparts in the special case where the weighted mean coincides with the arithmetic mean. The reduction of some results to ordinary geometry is also considered.

The current attempt is aimed to extend previous results, concerning the explicit expression of the arithmetic mean standard deviation distribution, to the general case of the weighted mean standard deviation distribution. To this respect, the integration domain is expressed in canonical form after a change of reference frame in the n-space, which is recognized as an infinitely thin n-cylindrical corona where the axis coincides with a coordinate axis and the orthogonal section is an infinitely thin, homotetic (n-1)-elliptical corona. The semiaxes are formulated in two different ways, namely in terms of (1) eigenvalues, via the eigenvalue equation, and (2) leading principal minors of the matrix of a quadratic form, via the Jacobi formulae. The distribution and related parameters have the same formal expression with respect to their counterparts in the special case where the weighted mean coincides with the arithmetic mean. The reduction of some results to ordinary geometry is also considered.

Cite this paper

Caimmi, R. (2015) The Weighted Mean Standard Deviation Distribution: A Geometrical Framework.*Applied Mathematics*, **6**, 520-546. doi: 10.4236/am.2015.63049.

Caimmi, R. (2015) The Weighted Mean Standard Deviation Distribution: A Geometrical Framework.

References

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[2] Greene, B. (1999) The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. W.W. Norton, New York.

[3] Caimmi, R. (2013) The Arithmetic Mean Standard Deviation Distribution: A Geometrical Framework. Applied Mathematics, 4, 1-10.

[4] Malkin, Z.M. (2013) On the Calculation of Mean-Weighted Value in Astronomy. Astronomy Reports, 57, 882-887.

http://dx.doi.org/10.1134/S1063772913110048

[5] Gnedenko, B. (1978) The Theory of Probability. Mir, Moscow.

[6] Caimmi, R. (2014) Il Problema della Misura. Aracne, Roma. (In Italian, In Press)

[7] Papoulis, A. (1965) Probabilities, Random Variables, and Stochastic Processes. McGraw-Hill, New York.

[8] Smirnov, V. (1970) Cours de Mathè matiques Supè rieures. Mir, Moscow.

[9] Gantmacher, F.R. (1966) Théorie des Matrices. Dunod, Paris.

[10] Spiegel, M.R. (1969) Mathematical Handbook of Formulas and Tables. Schaum’s Outline Series, McGraw-Hill, New York.

[1] Misner, C.W., Wheeler, J.A. and Thorne, K.S. (1973) Gravitation. W.H. Freeman & Company.

[2] Greene, B. (1999) The Elegant Universe: Superstrings, Hidden Dimensions, and the Quest for the Ultimate Theory. W.W. Norton, New York.

[3] Caimmi, R. (2013) The Arithmetic Mean Standard Deviation Distribution: A Geometrical Framework. Applied Mathematics, 4, 1-10.

[4] Malkin, Z.M. (2013) On the Calculation of Mean-Weighted Value in Astronomy. Astronomy Reports, 57, 882-887.

http://dx.doi.org/10.1134/S1063772913110048

[5] Gnedenko, B. (1978) The Theory of Probability. Mir, Moscow.

[6] Caimmi, R. (2014) Il Problema della Misura. Aracne, Roma. (In Italian, In Press)

[7] Papoulis, A. (1965) Probabilities, Random Variables, and Stochastic Processes. McGraw-Hill, New York.

[8] Smirnov, V. (1970) Cours de Mathè matiques Supè rieures. Mir, Moscow.

[9] Gantmacher, F.R. (1966) Théorie des Matrices. Dunod, Paris.

[10] Spiegel, M.R. (1969) Mathematical Handbook of Formulas and Tables. Schaum’s Outline Series, McGraw-Hill, New York.