The city of Megalopolis operates three sewage treatment plants in three different locations throughout the city. The daily proportion of operating capacity exhibited by the three plants can be represented as the outcome of a trivariate random variable with the following probability density function:

\(f\left(x_{1}, x_{2}, x_{3}ight)=\frac{1}{3}\left(x_{1}+2 x_{2}+3 x_{3}ight) \prod_{i=1}^{3} I_{(0,1)}\left(x_{i}ight)\), where \(x_{i}\) is the proportion of operating capacity exhibited by plant \(i, i=1,2\), 3 .

(a) What are the expected values of the capacity proportions for the three plants, i.e., what is \(\mathrm{E}\left[\begin{array}{l}X_{1} \\ X_{2} \\ X_{3}\end{array}ight]\) ?

(b) What is the expected value of the average proportion of operating capacity across all three plants, i.e., what is \(\mathrm{E}\left(\frac{1}{3} \sum_{i=1}^{3} X_{i}ight)\) ?

(c) Given that plant 3 operates at 90 percent of capacity, what are the expected values of the proportions of capacity for plants 1 and 2 ?

(d) If the daily capacities of plants 1 and 2 are \(100,000 \mathrm{gal}\) of sewage each, and if the capacity of plant three is \(250,000 \mathrm{gal}\), then what is the expected daily number of gallons of sewage treated by the city of Megalopolis?