1/2 
= t ky09 
s=7KR (12.157) 
e=7 KY 
and 
Kjkm 
Mn a, 12.158 
eS ea ee, ER) Core 
pi2= see | | exp |-izs— 25-23 
(27) 
-5\° + Se4 57+ 21.1108 § + 21015 § + 2o11S i| 
+ > Ajem(isy (is) Gs)" ds ds ds. (12.159) 
jkm 
Expanding the characteristics of the moments as shown in Eq. 12.159 into joint 
moments and modifying them into joint cumulants, Dalzell derived the expression for 
joint cumulants until the fourth degree from the modified frequency response character— 
istics H\(w), H2(W1@2), H3(@),W2W3), and functions of w by the same type of style as 
Eq. 12.119 for the case of p(z, z). Since these manipulations require many transactions 
involving the higher order terms in the expansions, he checked the order of magnitude of 
the functions, suggested the order to truncate the approximations and, by laborious 
manipulation utilizing Hermite polynomial, he obtained his approximation. 
From these expressions, arbitrarily denoting the standardized maxima or minima of 
response by v he finally derived the expression for p(v,0,z) using the cumulants up to 
the fourth order and then the probability distribution function for maxima and minima by 
0 
pew) == | [21 p(v, 0,2) az 
0 
== | Z p(v,0,z) dz (12.160) 
372 
