In statistics, a location family is a class of probability distributions parametrized by a scalar- or vector-valued parameter μ, which determines the "location" or shift of the distribution. Formally, this means that the probability density functions or probability mass functions in this class have the form
- fμ(x) = f(x − μ).
Here, μ is called the location parameter.
In other words, when you graph the function, the location parameter determines where the origin will be located. If μ is positive, the origin will be shifted to the right, and if μ is negative, it will be shifted to the left.
A location parameter can also be found in families having more than one parameter, such as location-scale families. In this case, the probability density function or probability mass function will have the form
- fμ,θ(x) = fθ(x − μ)
where μ is the location parameter, θ represents additional parameters, and fθ is a function of the additional parameters.
Additive noise
An alternative way of thinking of location families is through the concept of additive noise. If μ is an unknown constant and w is random noise with probability density f(w), then x = μ + w has probability density fμ(x) = f(x − μ) and is therefore a location family.