A Characterization of the Poisson Distribution and the Probability of Winning a Game

The probability P(λ, μ) that a team with mean score λ beats a team with mean score μ is calculated when the score of each team is Poisson distributed. It is found that ∂P(λ, μ)/∂λ is equal to the probability of a tie. When this equality holds for any distribution of the score of the team with mean μ, it is shown that the score of the team with mean λ must be Poisson distributed. The Poisson distribution is shown to fit certain baseball data, and it is also applied to some soccer data.