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Poisson Distribution. A Poisson random variable is the number of successes that result from a Poisson experiment. The probability distribution of a Poisson random variable is called a Poisson distribution. Given the mean number of successes (Î¼) that occur in a specified region, we can compute the Poisson probability based on the following formula:
where x is now an (n + 1)-dimensional vector consisting of n independent variables concatenated to a vector of ones. Here Î¸ is simply Î± concatenated to Î². Thus, when given a Poisson regression model Î¸ and an input vector x, the predicted mean of the associated Poisson distribution is given by. If Y i are independent observations with corresponding values x i of the predictor variables, then Î¸ can be estimated by maximum likelihood.