By Edward P. C.(Edward P.C. Kao) Kao

Meant for a calculus-based direction in stochastic techniques on the graduate or complicated undergraduate point, this article bargains a contemporary, utilized perspective.Instead of the normal formal and mathematically rigorous technique traditional for texts for this direction, Edward Kao emphasizes the improvement of operational talents and research via various well-chosen examples.

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T , o with ' ) covariance matrix is given as where IM,~ denotes the determinant of M,. 10), the joint moments are given as 18 PROBABILITY DISTRIBUTIONS INVOLVING GAUSSIAN RANDOM VARIABLES 10, k, + k2 odd k, + k2 even B. 10), then the joint PDF and CDF of R, and R2 are given by The joint moments of R, and R2 are given by where , F; (a,P;y ;x ) is the Gaussian hypergeometric function [2]. FUNDAMENTAL MULTIDIMENSIONAL VARIABLES 19 1. n = 2 Using the alternative representation of the first-order Marcum Qfunction given in Eqs.

1 (m,-l+i)! (m2- 1 - i)! (m, - I)! (m2- 1- i)! 2' where 2(0~: (*)I DIFFERENCE OF CHI-SQUARE RANDOM VARIABLES 33 is a generalization of the Marcum Q-function defined in [9, Eq. (86)] and which has recursive properties [9, Eq. 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.

B. Dependent Central Chi-square (-) Central ChiSquare DIFFERENCE OF CHI-SQUARE RANDOM VARIABLES 29 To simplify the expressions, we introduce the parameters 112 y- = [(of- o:)1+ 4o:of(l- p i ) ] o;o; (1- p 2 ) Note that a+2 0 and a-2 0. 19) o;0;2 (1- p 2 ) C. ; and Y, are independent noncentral and central chi-square distributed RVs with n, and n, degrees of freedom, respectively. 1 (m,-l+i)! (m2- 1 - i)! (m, - I)! (m2- 1- i)! 2' where 2(0~: (*)I DIFFERENCE OF CHI-SQUARE RANDOM VARIABLES 33 is a generalization of the Marcum Q-function defined in [9, Eq.