Xj = Xo +€X, 
E[X7] = E[X6] + 2€ E[XoX1] 
A(ty) A(t) Efft —t)flt—1)}dt, dr. (10.46) 
_— 8 
and E[X3] = | 
8 
When f(r) is stationary, 
RAt) = Efft) fit +7). 
For example, for Duffing type oscillations as 
X + 20X + wa[X + €X?] = fir), (10.47) 
in Eq. 10.40, 
g(X) = X?. (10.48) 
Therefore from Eq. 10.45, 
E[XoX1] = - 03 | h(t)E[Xo(0)X9(t —7) dt 
=- 0 | h(c)dt | h(ty)aty | h(t2)at | h(t3)at3 | h(t4)dta 
—c —o 
X Efft-—tp)ftt -—T-To)flt -T-T3) ft -—T-T,)]. 
When Z(r) is a zero mean stationary random process, the following relations apply: 
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