PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 323 



In each of these thice equations the very abbreviated notation at the ex- 

 treme right will be used wherever the function is being dealt with exten- 

 sively, as in the various succeeding sections. Such notation will not seem 

 unduly abbreviated nor arbitrary if the following considerations are noted: 

 In (1.10), «T]2 corresponds to the entire effective range of a, so that P(p \ o-]2) 

 is the 'principal' distribution function for p. Similarly, in (1.11), Q(< p,on) 

 is the 'principal' cumultive distribution function for p. In (1.12), the star 

 indicates that Q*ip) is the 'complementary' cumulative distribution func- 

 tion, since Q(p) + Q*(p) = Q(pi2 , 0-12) = 1, unity being taken as the measure 

 of certainty, of course. 



For occasional use in succeeding sections, the defining equations for 

 the probabiUty functions pertaining to four other particular cases will 

 be set down here: 



p{p<p' <P + dp, (Tx<a' <a) = P(p I < (t) dp, • (1.13) 



p(p< p' < p-{- dp, a < a' < (X2) ^ P(p \ > a) dp, (1.14) 



Pip, <p' <p,a,<a' <ct) = Q{< p, < a), (1.15) 



Pip <p' < p2 ,ai<a' <a) = Qi> p, < a). (1.16) 



It may be noted that (1.13) and (1.14) are mutually supplementary, in the 

 sense that their sum is (1.10). Similarly, (1.15) and (1.16) are mutually 

 supplementary, in the sense that their sum is ()(p]?,< a) = Qi< (r,pi2), 

 which is the correlative of (1.11). 



This section will be concluded with the following three simple trans- 

 formation relations (1.17), (1.18) and (1.19), which will be needed further 

 on. They pertain to the probability functions on the right sides of equa- 

 tions (1.3), (1.4) and (1.5) respectively, h and k denote any positive real 

 constants, the restriction to positive values serving to simplify matters 

 without being too restrictive for the needs of this paper. 



P{hp,ka) = ^^P{p,<t), (1.17) 



P{hp\k<rz,) =\Pip\ <^34), (1-18) 



Q{hpu,kazi) = Q{pzi, (T34). (1.19) 



Each of the three formulas (1.17), (1.18), (1.19) can be rather easily 

 derived in at least two ways that are very different from each other. One 

 way depends on probability inequality relations of the sort 



p{t<t'<t'Vdt) = p{gt<gt'<gt-^d[gt]), (1.20) 



p{h<t'<U) = p{gh<gl'<gh), (1.21) 



