52 BELL SYSTEM TECHNICAL JOURNAL 



and then draw a batch of a's. The probability that the curve obtained by- 

 putting these a's in (3.3-1) will have a zero in a;i , ici + dx is 



/+CO 

 I -n I P{^, v; xi) drj (3.3-2) 



00 



and the expected number of zeros in the interval (xi , X2) is 



^.v \r,\p(0,r,x)dr, (3.3-3) 



" Xl J— 00 



In these expressions p{^^ 77; x) is the probability density function for the 

 variables 



^ = F{ai , • • ay] x) 

 _ dF (3-3-4) 



dx 



Since the a's are random variables so are ^ and rj, and their distribution 

 will contain x as a parameter. This is indicated by the notation p(^, -q; x). 



These results may be proved in much the same manner as are similar 

 results for the distribution of the maxima of a random curve. This method 

 of proof suffers from the restriction that the a's are required to be bounded.'" 

 Results equivalent to (3.3-2) and (3.3-3) have been obtained independently 

 by M. Kac." His method of proof has the advantage of not requiring the 

 a's to be bounded. 



Here we shall sketch the derivation of a closely related result: The prob- 

 ability that y will pass through zero in .Vi , .Vi + dx wdth positive slope is 



dx / 77/'(0, 77; Xi) dr) (3.3-5) 



•'0 



We choose dx so small that the portions of all but a negligible fraction 

 of the possible random curves lying in the strip (xi , .Vi + dx) may be re- 

 garded as straight lines. If y = ^ at xi and passes through zero for .Vi < x < 



Xi -f- dx, its intercept on v = is Xi — - where rj is the slope. Thus ^ and rj 



must be of opposite sign and 



xi < xi — - < xi -f- dx 



V 



25 S. O. Rice, Amer. Jour. Math. Vol. 61, pp. 409-416 (1939). However, L. A. MacColl 

 has pointed out to me that a set of sufficient conditions for (3.3-5) to hold is: (a) pi^, v', x) 

 is continuous with respect to ($, 77) throughout the $77-plane; and (b) that the integral 



/ P(<in, v; Xi) dri 

 Jo 



converges uniformly with respect to a in some interval — ai < o < a^ , where ai and 02 

 are positive. These conditions are satisfied in all the applications we shall make use of 

 (3.3-5). 



2« M. Kac, Bjill. Amer. Math. Soc. Vol. 49, pp. 314-320 (1943). 



