Question: 58: The maintenance department in a factory claims that the number of breakdowns X of a particular machine follows a Poisson distribution with a mean of two breakdowns every 500 hours.

(a) Find \(\lambda \) and \({{\mu }_{X}}\)

(b) Write the formula for the exponential probability curve of X

(c) Sketch the probability curve

(d) Assuming that the maintenance department’s claim is true, find the probability that the time between successive breakdowns is at most five hours.

(e) Assuming that the maintenance department’s claim is true, find the probability that the time between successive breakdowns is between 100 and 300 hours.

(f) Suppose that the machine break down five hours after it most recent breakdown. Based on your answer to part (d), do you believe that the maintenance department’s claim is true?