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# Random Variables and Limit State Functions

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Consider the following two collections of data that represent realizations of two random variables X1 and X2:
X1: 18.9 21.1 17.8 20.2 16.0 19.0 20.9
19.1 22.5 18.7 15.:3 17.5 22.1 19.8 20.76
X2: 2:3.9 17.8 20.7 20.6 20.0 21.6 25.0
21.9 21.5 20.6 22.0 20.4 2:3.2 21.5 2:3.0
2:3.:3 21.8 2:3.8 26.6 2:3.0 22.0 2:3.8 22.1
(a) Estimate the mean and stadard deviation of each random variable.
(b) Assuiue that the two random variables are statistically independent and normally distributed.
i. Consider the limit state function L = X1 ? X2. Compute the probability of failure Pf = P[L > 0]. Is your computed answer exact or approximate?
ii. Now define the limit state function as L' = X1/X2 ? 1. Compute the probability of failure P'f = P[L' > 0] (hint: linearize L'). Is your computed answer exact or approximate?

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