Problem 1. A onestor}r building 9- with mass M, stiffness K, and damping C is modeled as a single degreeoffreedom oscillator. The natural frequency .0 and damping ratio Z of the oscillator are respectively dened by K C i]: d 2: VM 3\" 2m Suppose M , K, and C have the following secondmoment properties: Correlation Coefcients Variable (Units) Mean c.o.v K M C K (ka111) 500 0.33 1.0 0.3 0.0 M (stzfm) 2 0.25 0.3 1.0 0.0 C(stfm] 4 0.4 0.0 0.0 1.0 (a) Using rstorder approximations, compute the means and standard deviations of Q and Z, and the correlation coefcient between them. W'hich parameters contribute most to the uncertainties in Q. and Z? Problem 2. Let T1 and T2 denote respectively the times between successive arrivals; of automobiles and trucks at a highway stop. When the arrivals are entireiy random, one can show that T1 and T2 have the exponential distributions fin (t1) = \"EXP (031) $1 3* l} fa (t2) = exp (tz) a > 0 where oz and [3 are the respective mean arrival rates. Derive the joint PDF of the average and ratio of T1 and T2 T1+T2 = 1' fl Y: 2 m 3 T1 Use the joint PDF to derive an expression for the marginal PDF of Y2. Y1