PROBABILITY FUNCTIONS FOR COMPLEX VARIATE 321 



A generic quadrilateral region contained within the quadrilateral region 

 Pi , P2 , 0^1 , o'2 in Fig. 1.1 is the one bounded by arcs of the dashed curves 

 P3 , Pi , (T3 , (Ti , where ps < p4 and as < <j\ . Here, as in the preceding 

 paragraph, it will evidently suffice to deal with open inequalities. 



Referring to Fig. 1.1, the probability functions with which this paper 

 will chiefly deal are certain particular cases of the probability functions 

 P{p, a), P{p I 0-34) and Q{pz\ , C734) occurring on the right sides of the follow- 

 ing three equations respectively: 



p{p < p' < p ^ dp, (J < a' < a + d<r) = P(p,a)dpda, (1.3) 



p(p < p < p -^ dp, az < a' < (Ji) = P{p I (T3i)dp, (1.4) 



p{pz < p < Pi , (T3 < a' < (Ti) = Q{p3i , 0-34). (1.5) 



These equations serve to define the above-mentioned probability functions 

 occurring on the right sides in terms of the probabilities denoted by the 

 left sides, each expression p( ) on the left side denoting the probability 

 of the pair of inequalities within the parentheses. Inspection of these 

 equations shows that: P(p,(r) is the 'distribution function' for p and a 

 jointly; P{p \ 0-34) is a 'distribution function' for p individually, with the 

 understanding that a' is restricted to the range a^-to-ai ; Qipsi ,o'34) is a 

 'cumulative distribution function' for p and a jointly. 



Since the left sides of (1.3), (1.4) and (1.5) are necessarily positive, the 

 right sides must be also. Hence, as all of the probability functions occur- 

 ring in the right sides are of course desired to be positive, the differentials 

 dp and da must be taken as positive, if we are to avoid writing | dp \ and 

 I (/(T I in place of dp and da respectively. 



Returning to (1.3), it is seen that, stated in words, P{p,a) is such that 

 P{p.a)dpda gives the probability that the unknown values p' and a' of 

 the constituents of the unknown value r' of a random sample consisting 

 of a single r-variate lie respectively in the differential intervals dp and da 

 containing the constituent values p and a respectively. Thus, unless 

 dpda is the differential element of area, Pip,a) is not equal to the 'areal 

 probability density,' G{p,a), defined in the fourth paragraph of this section. 

 In general, if £ is such that Edpda is the differential element of area, then 

 P(p, a) = EG{p, a). (An illustration is afforded incidentally by Appendix A.) 



P{p,a), defined by (1.3), is the basic 'probabiUty function,' in the sense 

 that the others can be expressed in terms of it, by integration. Thus 



^ Thus p in p( ) may be read 'probability that' or 'probabiHty of.' 



