# Scale parameter

In probability theory and statistics, a scale parameter is a special kind of numerical parameter of a parametric family of probability distributions.

## Definition

If a family of probability densities with parameter s is of the form

$f_s(x) = f(x/s)/s, \!$

where f is a probability density function, then s is called a scale parameter, since its value determines the "scale" of the probability distribution. If s is large, then the distribution will be more spread out; if s is small then it will be more concentrated.

We can write $f_s$ in terms of $g(x) = x/s$, as follows:

$f_s(x) = f(x/s) \times 1/s = f(g(x)) \times g'(x). \!$

Because f is a probability density function, it integrates to unity:

$1 = \int_{-\infty}^{\infty} f(x)\,dx = \int_{g(-\infty)}^{g(\infty)} f(x)\,dx. \!$


By the substitution rule of integral calculus, we then have

$1 = \int_{-\infty}^{\infty} f(g(x)) \times g'(x)\,dx = \int_{-\infty}^{\infty} f_s(x)\,dx. \!$


So $f_s$ is also properly normalized.

## Rate parameter

Some families of distributions use a rate parameter which is simply the reciprocal of the scale parameter. So for example the exponential distributions with scale parameter β and probability density

$f(x;\beta ) = \frac{1}{\beta} e^{-x/\beta} ,\; x \ge 0$

could equally be written with rate parameter λ as

$f(x;\lambda) = \lambda e^{-\lambda x} ,\; x \ge 0.$

## Examples

• The normal distribution has two parameters: a location parameter $\mu$ and a scale parameter $\sigma$. In practice the normal distribution is often parameterized in terms of the squared scale $\sigma^2$, which corresponds to the variance of the distribution.
• The gamma distribution is usually parameterized in terms of a scale parameter $\theta$ or its inverse.
• Special cases of distributions where the scale parameter equals unity may be called "standard" under certain conditions. For example, if the location parameter equals zero and the scale parameter equals one, the normal distribution is known as the standard normal distribution, and the Cauchy distribution as the standard Cauchy distribution.