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Showing posts from May, 2017

Solving IMO 1989 #6 using Probability and Expectation

Poisson Games and Sudden-Death Overtime

Let's say we have a game that can be reasonably modeled as two independent Poisson processes with team \(i\) having parameter \(\lambda_i\). If one team wins in regulation with team \(i\) scoring \(n_i\), then it's well-known we have the MLE estimate \(\hat{\lambda_i}=n_i\). But what if the game ends in a tie in regulation with each team scoring \(n\) goals and we have sudden-death overtime? How does this affect the MLE estimate for the winning and losing teams?

Assuming without loss of generality that team \(1\) is the winner in sudden-death overtime. As we have two independent Poisson processes, the probability of this occurring is \(\frac{\lambda_1}{\lambda_1 + \lambda_2}\). Thus, the overall likelihood we'd like to maximize is \[L = e^{-\lambda_1} \frac{{\lambda_1}^n}{n!} e^{-\lambda_2} \frac{{\lambda_2}^n}{n!} \frac{\lambda_1}{\lambda_1 + \lambda_2}.\] Letting \(l = \log{L}\) we get \[l = -{\lambda_1} + n \log{\lambda_1} - {\lambda_2} + n \log{\lambda_2} - 2 \log{n!} …