MATHEMATICAL ANALYSIS OF RANDOM NOISE 53 



According to the statement of our problem, we are interested only in positive 

 values of rj, and we therefore write our inequality as 



-vdx < ^ <0 



For a given random curve i.e. for a given set of a's ^ and rj have the values 

 given by 



^ = F(ai , • • • a.v ; Xi) 



D 



' ~ '.^i=x. 



If these values of ^ and rj satisfy our inequality, the curve goes through zero 

 in Xi , Xi + dx. The probability of this happening is 



[ drj [ d^p{^,n;xi) = [ [0 - (-rj dx)]p(0, r,; Xi) d:j 



Jo J-v dx Jl\ 



where we have made use of the fact that dx is so very small that ^ is cVcc- 

 tively zero. The last expression is the same as (3.3-5). 



In the same way it may be shown that the probability of y passing through 

 zero in a'l , Xi + dx with a negative slope is 



—dx I vp{^, V, Xi) dr] (3.3-6) 



Expression (3.3-2) is obtained by adding (3.3-5) and (3.3-6). 



We are now ready to apply our formulas. We let /, I(t) and (pn play the 

 roles of X, y, and c„ , respectively, and use 



lii) = S Cn COS {(^J - ^n), c~n = 2w(f)Af (2.8-6) 



n=l 



2' MacColl has remarked that the step from the double integral on the left hand side 

 of this equation to the final result (3.3-5) may be made as follows: 

 It is easily seen that the proliability density we are seeking is 



77—. / dv / Pi^, r, x) d^ 



Proceeding formally, without regard to conditions validating the analytical operations 

 (for such conditions see the footnote on page 52), we have 



-JT I dr) \ pii, 7}-, x) d^ = I riPi-V'^x, r, x) 



dAx Jj J_,^^ Jo 



and hence the required probability density is 



ripiO, r);x)dri 



J 



Jo 



