The maintenance department in a factory claims that the number of breakdowns of a particular machine follows a Poisson distribution with a mean oftwo breakdowns every 500 hours. Let Xdenote the time (in hours) between successive breakdowns. (a) Find A and ,ux. (Write the fraction in reduced form.) (b) Write the formula for the exponential probability curve of X. (d) Assuming that the maintenance department's claim is true, find the probability that the time between successive breakdowns is at most five hours. (Round your answer to 4 decimal places.) P (x s 5) 0.0198 (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. (Round your answer to 4 decimal places.) P (100 S x S 300) 0.3691 (f) Suppose that the machine breaks down five hours after its most recent breakdown. Based on your answer to part 0', do you believe the maintenance department's claim? (Round your answer to 4 decimal places.) \"lll1 ; the probability of this result is quite