Suppose you visit a province where license plates numbers are numbered sequentially. After seeing some cars go by on the road and reading their license plate numbers, you wonder: how many cars in total are there? What might the next number I see be? To formalize the problem, we assume that all cars in the province are numbered from 1 to L, where I is the largest licence plate number. Let M be the largest possible value of L. To make things simple, we'll assume that license plate numbers are three digits, so that M = 900. We assume that all values of [ are equally likely, so our prior for L is a uniform distribution from 1 to M. Furthermore, we assume that, when we see a new car, we are equally likely to see any of the [ cars out there, so the likelihood of seeing licence plate number X is also uniform. Our observations will be the numbers Xi of the NV cars we see go by. To specify the model, we define F(Z, A. D) = B-All AcZEB 0 otherwise (1) P(L) = f(L, 1, M) (the prior) (2) P(X]L) = f(X, 1, L) (the likelihood of a single license plate number X") (3) N P(XIN|[) = [P(Xi|L) (the likelihood of observing numbers XI:N) (4) il Additionally, define X max = max XIN (5) to be the largest license plate number observed. (a) Write the posterior distribution P(L|X:w) using Bayes' Rule, in terms of the uniform distributions above. Hints: simplify the numerator first. For the denominator, use the sum rule and the product rule: P(X(:N) = CM, P(XIN, L =1) = EM, P(Xi:|L =1)P(L =i). How does the denominator relate to the numerator? (b) For what range of values of [ will the posterior be non-zero? (c) For the values of [ where the posterior is nonzero, simplify the posterior into a function of only L, M. N, Xmax, and or the observations. Hint: begin with the numerator