Problem 2. Consider the site located at 50 km distance from one end of a 200 km long fault. An earthquake on the fault may occur as a rupture of length S, where S is a random variable with PDF fs (s) = 10 kms 100 km $2 The rupture may occur anywhere within the fault with equal likelihood. 50 km site R 200 km rupture. www fault S (a) Determine the parameter k. (b) Derive the conditional CDF and PDF of the shortest distance from the structure to the rupture, denoted R, given that S = s. Sketch the conditional PDF of R for s = 50 km. (c) Derive the joint PDF of R and S. Make a qualitative 3D sketch of this PDF. (d) Determine the numerical values of the following probabilities: 1 Department of Civil and Environmental Engineering University of Illinois at Urbana-Champaign CEE 491 Decision and Risk Analysis Instructor Paolo Gardoni P (R75 km S = 50 km) P (R75 km) + In (a+bu). You may use the relation (a+b) du au