1 - Piecewise density for waiting time. A support engineer models the waiting time T (in minutes) for a customer service call by the piecewise pdf [ f_T(t) = k,t, & 0 t < 2, k,(4 - t), & 2 t 4, 0, & otherwise, where k is a constant. (a) Determine the constant k so that f_T integrates to 1. Sketch the pdf and describe its shape. (b) Find the cumulative distribution function F_T and evaluate P(1 < T < 3). Show that P(T=2) = 0 because the probability at a point is zero for a continuous distribution. (c) Compute the expected value E[T] and variance Var(T) by integrating over the two regions. Interpret the mean waiting time and the spread in the context of customer service. (d) Management wishes to guarantee that 95 % of customers wait no more than t_95 minutes. Solve for t_95 using the CDF. Discuss whether the target is reasonable given the mean and variance