(c) At a bus stop, there are two buses available (bus numbers 1 and 2). For i = 1, 2, the waiting time until bus i arrives is denoted by T;, with T independent of T2, and T; has density function fr. (t) = X, exp(-1;t), t>0. You are waiting for either bus 1 or 2 to arrive. Let W denote the waiting time until you can get on a bus. i. Explain briefly why W min(T, T). ii. Show that E(W) < 1/max(\1, A2). (Hint: You need to work out E(T¡) and E(T,) first.) ii. Work out the probability density function of W. (Hint: Find P(W > w) first for w > 0.)

The following table represents collected data from literature regarding the inelastic axial capacity (Pn) of Aluminum symmetric profiles as a function of the Area of the section and the Slenderness Ratio (SR) of the bar. Produce a linear model for the capacity in the form Pn= C1 X Area + C2 X SR, show the statistics of the linear model on this paper. Use your linear model to predict Pn for the case of Area= 200 and SR = 35. Produce a 95% confidence interval for your expected value of Pn at Area=200mm2 and SR=35. Find the maximum possible Pn and the minimum possible Pn based on 95% confidence limits of the parameters. Suggest how to improve your model by saying which one of the two factors must be treated nonlinearly and show why you think so! (give at least one direct reason) Area (mm2) SR (-) Pn (kN) 198.3 16 1273 216.3 89 136.5 228 94 129 184.3 83 133.8 220.1 99 112.3 166.7 33 765.4 201.5 70 205.6 219.8 53…