A system consists of three independent components X, Y, and Z whose lifetimes are exponentially distributed with means 1/λX, 1/λY and 1/λZ, respectively. Determine the PDF and expected value of W, the time until the system fails (or the lifetime of the system), under the following system configurations:
a. The components are connected in series
b. The components are connected in parallel
c. The components are connected in a backup mode with X used first, then Y, and then Z.