The number of guests staying in a hotel in a small town is a Poisson random variable with parameter A = 10, i.e., N Pois(A). The hotel management needs to provide enough hot water to be used in hotel rooms. Each guest i, consumes W; gallons of hot water each night, where W; is an exponential random variable with E[W] = 7 gallons. We assume that the water consumption of guest i is independent from that of guest j. (a) Let X be the total consumption of hot water in the hotel in a random day. Write X in terms of N and W₁'s. (b) Find the mean and variance of X. (c) The hotel management want to make sure that they can provide enough hot water for their guests with probability at least 95%. Use Chebyshev's inequality and determine the minimum amount of hot water required to satisfy the management's goal. (d) Approximate X by a Gaussian random variable, and determine the minimum amount of hot water needed to guarantee that the chance of running out of hot water is at most 5%.