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# Suppose that Batman will have n battles next year (with n a positive integer), and that each battle is with the Joker with probability p1, Catwoman

Suppose that Batman will have n battles next year (with n a positive integer), and that each battle is with the Joker with probability p1, Catwoman with probability p2, and the Riddler with probability p3, independently. Here p1, p2, p3 are nonnegative and sum to 1. Let X1, X2, X3 be the numbers of battles Batman will have with the Joker, Catwoman, and the Riddler next year, respectively

) Suppose for this part only that the parameters p1, p2, p3 are unknown, n = 360, and it is observed that exactly 36 of the battles are with the Riddler. A natural way to estimate p3 would be to use 36/360 = 0.1. The maximum likelihood estimate (MLE) of p3 is the value of p3 that makes the observed data, X3 = 36, as likely as possible. That is, the MLE is the value of p3 that maximizes P(X3 = 36). Show that the MLE is the natural estimate, 0.1. Hint: Take the log before maximizing.