generalized linear model
Given a random vector, or the response variable, Y, a generalized linear model, or GLM for short, is a statistical model $\{{f}_{\text{\mathbf{Y}}}(\bm{y}\mid \bm{\theta})\}$ such that

1.
the components^{} of Y are mutually independent of each other,

2.
${f}_{{Y}_{i}}({y}_{i}\mid {\theta}_{i})$ belongs to the exponential family of distributions and has the following canonical form:
$${f}_{{Y}_{i}}({y}_{i}\mid {\theta}_{i})=\mathrm{exp}[y{\theta}_{i}b({\theta}_{i})+c(y)],$$ where the parameter ${\theta}_{i}$ is called the canonical parameter and $b({\theta}_{i})$ is called the cumulant function.

3.
for each component or variate ${Y}_{i}$, with a corresponding set of $p$ covariates ${X}_{ij}$, there exists a monotone^{} differentiable function $g$, called the link function, such that
$$g(\mathrm{E}[{Y}_{i}])=\text{\mathbf{X}}_{i}{}^{\mathrm{T}}\bm{\beta},$$ where $\text{\mathbf{X}}_{i}{}^{\mathrm{T}}=({X}_{i1},\mathrm{\dots},{X}_{ip})$, and $\bm{\beta}={({\beta}_{1},\mathrm{\dots},{\beta}_{p})}^{\mathrm{T}}$ is a parameter vector.
In practice, an extra parameter called the dispersion parameter, $\varphi $, is introducted to the model to lower a phenonmenon known as overdispersion. The GLM now looks like:
$${f}_{{Y}_{i}}({y}_{i}\mid {\theta}_{i})=\mathrm{exp}[\frac{y{\theta}_{i}b({\theta}_{i})}{a(\varphi )}+c(y,\varphi )]$$ 
Remarks

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Below is a table of canonical parameters and cumulant functions for some wellknown distributions from the exponential family:
Title generalized linear model Canonical name GeneralizedLinearModel Date of creation 20130322 14:30:11 Last modified on 20130322 14:30:11 Owner CWoo (3771) Last modified by CWoo (3771) Numerical id 17 Author CWoo (3771) Entry type Definition Classification msc 62J12 Synonym GLM Defines link function Defines canonical parameter Defines cumulant function Defines variance function \@unrecurse