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Probability distributions involving Gaussian random by Marvin K. Simon

By Marvin K. Simon

Likelihood Distributions related to Gaussian Random Variables: A instruction manual for Engineers and Scientists brings jointly an unlimited and finished choice of mathematical fabric in a single place, in addition to supplying a number of new effects interpreted in a kind that's quite invaluable to engineers and scientists.

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A single-sided Gaussian RV with PDF and CDF given by Also, the moments of R are given by 10 PROBABILITY DISTRIBUTIONS INVOLVING GAUSSIAN RANDOM VARIABLES E { R ~= J (202r'2 (F), k integer Here R corresponds to a conventional Rayleigh RV with PDF and CDF Also, the moments of R are given by ( 3 r(rn+ E) E { R ~=) (202)*i2T 1+ - ,k integer E { R ~=} (202)*i2 ( m - l)! 13) C. Rician where , ~ ; ( a ; P ; y is ) the confluent hypergeometric function [2] and a= 1x1. 2 where is the generalized (mth-order)Marcum Q-function [8].

Because of the absence of the functions in standard mathematical software manipulation packages such as MathematicaG9, and the complexity of the resulting expressions, their use is somewhat limited in practical applications and thus the author has decided to omit these results for this case. Nevertheless, the CF is still simple and given by D. Independent Central Chi-square (-) Noncentral Chi-square Define Z = Y; - Y, where Y; and Y, are independent central and noncentral chi-square distributed RVs with n, and n, degrees of freedom, respectively.

24) by Before concluding this section we point out that for the case n, = n, = m, m odd, the PDF can be expressed in the form of an infinite series in Whittaker functions [2] which themselves are expressed in terms of the confluent hypergeometric function , 4(a;P;y ) . Because of the absence of the functions in standard mathematical software manipulation packages such as MathematicaG9, and the complexity of the resulting expressions, their use is somewhat limited in practical applications and thus the author has decided to omit these results for this case.

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