5. Reliability of a computer storage system (15 points). Consider a hierarchical storage system composed of n subsystems, each comprising k servers. Each subsystem can tolerate a single server failure, and the overall system can tolerate a single subsystem failure. Thus, in order for the overall system to fail, there has to be at least two sub systems that each have at least two server failures. Suppose servers fail independently with probability p. (a) Let X be the number of servers that failed in a particular subsystem. Find the pmf of X in terms of k and p. What is the probability 190 that a particular subsystem fails? (b) Let Y be the number of subsystems that failed. Find the pmf of Y in terms of n and 1%. What is the probability 191 that the overall system fails? (0) Give upper bounds on p0 and 191 using the union of events bound. Hint: Let Aij be the event both server i and j fail for 1 S i