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This page provides some help in setting probability function paramaters
A Perfect Probability Function?
There is no objectively perfect model of the probability distributions of game scores between any two teams, and there are tons of choices for functions to represent probability distributions. This is where we must choose some approximate parameterized function to represent the range of probability distributions of possible game scores. Our goal is a probability distribution function that can capture a reasonable range of match-ups, and which is reasonably easy for an expert to express.
For psychological reasons involving motivation - in a real match game there may be some more complex interaction between team scores in a match, but this seems unlikely to have a significant effect on which team wins the game.
Truncated Gaussian and Poisson Distributions
To represent the probability distribution of scores in a game, we use either a Truncated Gaussian or a Poisson distribution, with user specified means (and standard deviations in the case of a Gaussian). For each possible game match-up (i.e. every pair of teams), an expert must provide an estimate of the mean number of goals scored by each team. The mean values can be seen as a kind of average score if the two teams played over and over again.
Standard Deviations
If using Truncated Gaussian functions for the distribution of scores, an expert also needs to provide an estimate of the Standard Deviation of the provided means. The standard deviation can be seen as a measure of the noise in the estimate. More precisely, it is a measure of the average distance of the data values from their mean. The larger the standard deviation, the more randomness that will result when the games are simulated. With standard deviations of 0, the expert is saying that he is 100% confident of the exact score of the game. Larger values reflect less confidence (or more innate randomness). Standard deviation parameters are ignored when using Poisson distribution.
Which probability Function Should You Use?
You can choose which probability distribution to use. In practice we don't expect much difference between using a Gaussian and Poisson distribution. The Poisson is a bit simpler since it does not require a Standard Deviation value to be provided.