In some families of distributions, a linear transformation results in another distribution within the same family.

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 location parameter is affected by adding a constant to \(X\); a scale parameter is affected by multiplying \(X\) 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.

We now apply this to normal distributions.

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 \(b\sigma\).