Problem 5. A system suffers faults by a Poisson process with rate 1. Each fault will, independently, cause the system to fail after time X with c.d.f. F. When the system fails, the responsible fault is fixed, but earlier other faults may remain and cause later failure. (a) What is the distribution of the number of failures up to t? (b) What is the distribution of the number of faults that remain in the system at time t and are not yet fixed? (c) Are the random variables in parts (a) and (b) dependent or independent? (justify!)