Also, the expected number of mean level up—crossings is 
loo} 
1 
EIN.) = 5 | lzo| p(Z, 22)dz2. (12.116) 
The probability distribution function of a maxima then lies above Z; is 
E[(N4(z 
P(z1) = ay (12.117) 
E[N.(21)] 
Substituting Eq. 12.113 into Eq. 12.115 and carrying on the integration gives 
1 [kg Ze 
E[(N.(z)] = — foo exp _———— 
2m 20 2k20 
ko Zz £ ko Z1 
S| apa 18 hy || SS ea 
vk20 5 kao Vk20 Vk20 
kK 
ee Ay hae) 
€ 
+> oo 
2 korVkoo  \ Vka0 
As already shown in Eq. 12.111, the cumulants are calculated from the m,nth 
moment of response, and the m,nth moment can be expressed by using the frequency 
functions and the input spectrum as some of the examples that were shown in Section 
11.4. With the order of each term up to 0 €) taken into account, and with the spectrum 
function, which is a real function, the cumulants can be written as follows: 
359 
