Variance unconditional

Note that, like the unconditional variance, the conditional variance can be expressed as... 

For this reason, mixing in the conditional PDF model is orthogonal to mixing in the unconditional model. Thus, y in the conditional model need not be the same as in the unconditional model, where its value controls tiie mixture-fraction-variance decay rate. 

For the second step, we need an estimate of a0, which is the unconditional variance if there are no ARCH effects. We computed this based on the ARCH specification by a regression of et2 - (8/36)et j2 -. .. -(l/36)et s2 on just a constant term. This produces a negative estimate of do, but this is not the variance, so we retain the result. We note, the problem that this reflects is probably the specific, doubtless unduly restrictive. ARCH structure assumed. 

Let also n0 be replicates of the above experiment where the hazard rate varies from experiment to experiment with probability density function / (h). From the previous relations, the unconditional expectation and variance are (cf. Appendix D)... 

The prediction-error variance in all cases is significantly reduced from the unconditional variance of xi (i.e., 1.0) because even the lowest correlation coefficient (between x and X4) is —0.95 —0.8574, which is considered negatively highly correlated. Of course, it is expected that using X2 yields a prediction with the smallest uncertainty. 

Further, according to (13), the variances and standard deviations of the system unconditional lifetimes in the system reUabihty state subsets are... 

In order to evaluate the accuracy of this approach, the method has been applied to a simple random vibration problem, namely the estimation of the spectrum standard deviation of the displacement of a SDOF linear system with damping ratio = 0.05. To this end use is made of the total probability theorem, applied to estimate the unconditional variance of the structural displacement d 6), considered as a function of the ground motion parameters 6 = cwg, Vg, Ag, as follows ... 

Taking into account that the mean of the random displacement is zero, the unconditional variance of the response, given by Eq. (67), is estimated with the point estimate technique as... 

If the integral in Eq. (5.25) is calculated with a fixed value of a single parameter Xj, then the variance caused by all other parameters except for Xj denoted by V(J, xj) is obtained. If this V(Ji xj) value is calculated for many values of Xj, selected according to its pdf, then the expected value E(y(Xi xj)) can be calculated. This requires the integration of V(Yi Xj) over the pt oi Xj (see Saltelli (2002) for details). The value V(Yi)-E(V(Yi Xj)) is equal to the reduced variance of F, caused by fixing the value of Xj and is equal to V E(Yi xj)). Dividing this conditional variance by the unconditional variance, the first-order sensitivity index for parameter Xj can be calculated ... 

For independent inputs (i.e. no correlations exist between inputs), a unique decomposition of the unconditional variance V Y) which parallels the decomposition given in Eq. (5.41) can be obtained (Li et al. 2010) ... 

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