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
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
where μ is the location parameter, θ represents additional parameters, and fθ is a function of the additional parameters.
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.
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