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| rowspan=2| [[categorical distribution|Categorical]]
| integer: <math>[0,K)</math>|| rowspan=2| outcome of single ''K''-way occurrence
| rowspan="3" |<math>\mathbf{X}\boldsymbol{\beta}=\ln \left(\frac \mu {1-\mu}\right) \,\!</math>
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| ''K''-vector of integer: <math>[0,1]</math>, where exactly one element in the vector has the value 1
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| [[multinomial distribution|Multinomial]]
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In the cases of the exponential and gamma distributions, the ___domain of the canonical link function is not the same as the permitted range of the mean. In particular, the linear predictor may be positive, which would give an impossible negative mean. When maximizing the likelihood, precautions must be taken to avoid this. An alternative is to use a noncanonical link function.
In the case of the Bernoulli, binomial, categorical and multinomial distributions, the support of the distributions is not the same type of data as the parameter being predicted. In all of these cases, the predicted parameter is one or more probabilities, i.e. real numbers in the range <math>[0,1]</math>. The resulting model is known as ''[[logistic regression]]'' (or ''[[multinomial logistic regression]]'' in the case that ''K''-way rather than binary values are being predicted).
For the Bernoulli and binomial distributions, the parameter is a single probability, indicating the likelihood of occurrence of a single event. The Bernoulli still satisfies the basic condition of the generalized linear model in that, even though a single outcome will always be either 0 or 1, the ''[[expected value]]'' will nonetheless be a real-valued probability, i.e. the probability of occurrence of a "yes" (or 1) outcome. Similarly, in a binomial distribution, the expected value is ''Np'', i.e. the expected proportion of "yes" outcomes will be the probability to be predicted.
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