Stats problem below.

You are responsible for reliability with a large equipment manufacturer. There are 72 large trucks in your machine park. Assume each one has a probability of breakdown of 0.1 for a given day. Breakdowns are independent of one another. Let X be the number of breakdowns.

• How is X distributed?
• What is the probability of getting 7 breakdowns on a given day?
• What is the mean number of breakdowns on a given day?
• What is the standard deviation of X?
• Is it reasonable to use a normal probability approximation to this problem? Why or why not?
• Assume it is reasonable to use a normal probability approximation, and that you use the answers in c and d as your mean and standard deviation for a normal distribution approximation to the distribution of X. What does this distribution say is the probability of getting at least 9 breakdowns on a given day?

This question builds off of above and all values from above also apply to this question. At the weekly level, repairs are conducted so quickly that you believe it is justified to claim that what happens in any one interval of time, e.g. 1 day or 4 weeks, is independent of what happens in any other interval, and that the distribution of breakdowns is the same for any two non-overlapping intervals of equal length. Let Y be the number of breakdowns per week. Assume that the expected value of Y, E(Y), is 62.

• How is Y distributed?
• What is the probability of getting precisely 62 breakdowns in a given week?
• Let W be the wait between breakdowns. How is W distributed?
• What is the probability that the time between the next two breakdowns is less than 3 hours?
• What is the probability that the time between the next two breakdowns is exactly 3 hours?
• What is the probability that more than 10 hours pass between the next two breakdowns?
• Split the next month into 30 day shifts and 30 night shifts, each of 12 hours. Note that work continues without let up for day or night.

i. What is the probability of getting exactly one such shift in the next month without a breakdown?

ii. What is the probability of getting more than one shift without a breakdown?