In some families of distributions, a linear transformation results in another distribution within the same family, often with only a single parameter changing. In the family, a location parameter is one that is affected by adding a constant.

Definition

In a family of distributions, \(X \sim \mathcal{Distn}(\theta)\), the parameter \(\theta\) is called a location parameter if \(Y = (X + a) \sim \mathcal{Distn}(\theta + a)\).

If the family of distributions has additional parameters, they should remain unchanged after the transformation.

A scale parameter can be similarly defined as one that is affected by multiplying the variable by a constant.

Definition

In a family of distributions, \(X \sim \mathcal{Distn(\phi)}\), the parameter \(\phi\) is called a scale parameter if \(Y = bX \sim \mathcal{Distn(b\phi)}\).

In families of distributions with a location parameter \(\theta\), \(X \sim \mathcal{Distn(\theta, \phi)}\), the parameter \(\phi\) is also called a scale parameter if \(Y = bX \sim \mathcal{Distn(b\theta, b\phi)}\).

If the family of distributions has additional parameters, they should again remain unchanged after the transformation.

From the results on the previous page, the parameters \(\mu\) and \(\sigma\) of a \(\NormalDistn(\mu, \sigma^2)\) distribution are location and scale parameters.

Normal distribution

If \(X \sim \NormalDistn(\mu, \sigma)\), we showed that \(Y = a + bX \sim \NormalDistn\left(a + b\mu, (b\sigma)^2\right)\).

Since \(Y = X + a \sim \NormalDistn\left(a + \mu, {\sigma}^2\right)\), the distribution's first parameter, \(\mu\), is a location parameter.

\(\sigma\) satisfies the second definition for scale parameters since if \(X\) has a normal distribution with parameters \(\mu\) and \(\sigma\), \(bX\) has one with the corresponding parameters \(b\mu\) and \(b\sigma\).