Problem 1. Consider the site located at 50 km distance from one end of a 200 kn] long fault. An earthquake on the fault may occur as a rupture of length 8, where S is a random variable with PDF fats) = g; lUkisg 101} km The rupture may occur anywhere within the fault with equal likelihood. 20!] km rupture l I I fault S If. site (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 = 3. Sketch the conditional PDF of R for 5 = 50 km. (c) Derive the joint PDF of R and 8. Make a qualitative 3D sketch of this PDF. (d) Determine the numerical values of the following probabilities: P(R E 75km | S = 50km) was 75km) You may use the relation If 3% = i + Eb; ln (5%)