# Introduction¶

The single-parameter exponential family is a class of distributions that can be expressed as:

```
f(y; θ, 𝜙) = exp{(yθ - b(θ))/a(𝜙) + c(y,𝜙)}.
```

for which `𝜙`

is assumed to be known.
The definition of the functions `a(.)`

, `b(.)`

, and `c(.)`

determines a
probabilistic distribution having the canonical parameter `θ`

. The expectation of
`y`

, denoted here by `𝜇`

, determines the value of `θ`

via the following relation:

```
b'(θ) = 𝜇
```

Still, the value `𝜇`

is often set indirectly via the natural parameter `η`

, which
relates to each other through a link function `g(.)`

:

```
η = g(𝜇)
```

If `g(.)`

is the so-called canonical function, we have the desirable equality:

```
θ = η
```