6.58 The maintenance department in a factory claims that the number of breakdowns of a particular machine follows a Poisson distribution with a mean of two breakdowns every 500 hours. Letxdenote the time (in hours) between successive breakdowns.

a FindAandmx.

b Write the formula for the exponential probability curve ofx.

c Sketch the probability curve.

d Assuming that the maintenance departments claim is true, find the probability that the time between successive breakdowns is at most five hours.

e Assuming that the maintenance departments claim is true, find the probability that the time between successive breakdowns is between 100 and 300 hours.

f Suppose that the machine breaks down five hours after its most recent breakdown. Based on your answer to partd, do you believe the maintenance departments claim? Explain.