• The linear component ηi, de?ned just in the traditional way: η_i = x'₁
• A monotone differentiable link function g that describes how E(Yi) = µi is related to the linear component:

• Yi are independent (i = 1, 2, . . . , n) (but not identically distributed) and each has a distribution from the exponential family. For the purpose of Generalized Linear Models theory, the usual exponential family is parametrized in a bit more speci?c way. Let us focus for simplicity to the case where θ is one-dimensional. It is assumed that the probability density of the response Y can be expressed as

for some functions a_i(Φ) = Φ/wi, b and c_i = c(y_i, Φ/wi), where wi is a known weight for each observation. Hereby, θ_i is called canonical parameter whereas Φ is called the dispersion parameter.

Note: For most models, the weight is the same for each i (that is, w_i = 1 for all i) and then the scaling simpli?es to a_i(Φ) = Φ), whereas c_i(y_i, Φ) becomes just c(y_i, Φ) and (1) simpli?es to

It can be shown from a standard theory for these distributions that