An Application of the Bayes Classifier: The following is a (very) simplified application of Bayes Theorem to business intelligence. In World War II the allied estimated the number of Germany’s tanks simply by looking at the serial number of those tanks they destroyed in the battle. Similar approaches have been used by businesses to estimate resources of their competitors. Here we give a simple example to understand this approach. Suppose a railroad company numbers its trains sequentially from 1001 to 1000+N, where N is the number of trains it has. The competing railroad firm does not know N and wants to estimate it. They send one of their employees, Sara, to do that. a. On the first day Sara observes the tracks their competition runs their trains on. She observes that a train numbered 1074 passes by. Assume that each of trains was equally likely to have passed by. For prior, assume that we estimate that there is no way the competition has more than 500 trains, and that, absent any information, any number N of trains up to 500 is equally likely. Based on this single observation, and using the Bayes decision rule, what is the best guess of N, the number of trains the company has? Show all your work, and then, to answer this question, use R to calculate and plot the posteriors. Also, calculate the expected value of N given Sara’s observation. b. Sara spent a few more days, and noticed that, in addition to train number 1074, trains number 1055, 1033, 1010 also passed through. Based on this new information, what is the best estimate of the number of trains based on the Bayes decision rule? And what is the expected value of N given all observations?