Suppose your company is selling a machine that contains 10 working components. When all components are working, each component has an exponentially distributed longevity with a mean of 50 years. If any of these components break and the total number of components reduces to 9, then the longevity of each remaining component is reduced to a mean of 40 years. If the number of functioning components reduces to 8, then the longevity of each remaining component is further reduced to a mean of 10 years. The machine breaks down if there are fewer than 8 functioning components. Assume that the machine costs $100,000. The manufacturer decides that an average refund of $10,000 per machine is acceptable. Your company wants to have a policy in that pays the buyer a full refund if it fails before year X, and a 50% refund if it fails before year Y. Your job is to determine appropriate values for X and Y. Here is a suggested outline: a) Give an overview of the problem to be solved. Also give a background of the exponential distribution and its connection to the Poisson distribution. b) Analyze the overall longevity of this machine. Include a discussion of your Matlab algorithm (don't just paste the code, discuss what the algorithm does). Discuss the longevities by providing a histogram and giving the average longevity. c) Modify your code so that each simulation gives the total amount spent on refunds. Experiment with different values of X and Y, keeping in mind that you want an average refund of $10,000. Note that the value of Y will depend on X. d) Give an overall recommendation to your supervisor with at least two options for X and Y. Coding hints: There are several ways to implement this - I recommend using the code for generating Poisson random variables, but instead of using the stopping condition "time < 1" you need to stop once the number of working components reaches a certain value. You need to put the code inside of a simulation loop and keep track of the overall time required for the machine to break down. You will also need to keep track of the refund for each simulation.