For the following performance function, determine an estimated failure probability and its COV using

(a) the direct simulation method, and

(b) the CE method with 100 and 2000 cycles:

The last term in this equation introduces noise to the performance function. Assume that the constants I1 = 1 and I2 = 1. The noncorrelated random variables are assumed to have the following probabilistic characteristics, with W having a standard deviation of 1:

Random Variable Mean Value COV Distribution Type X1 10 0.25 Normal X2 5 0.05 Normal X3 3 0.20 Normal W 0 Not applicable

(σ = 1)

Standard Normal

Z = X - X-X, +1, sin (w)