Parameter Types

Introduction to Parameter Types

Recommended Prerequesites

Definition

In probability theory and statistics, distributions are characterized by parameters that define their shape, spread, location, and other properties. This chapter will explain three key types of parameters commonly associated with probability distributions: location (loc), scale, and shape. The letter used to denote each depend on the distribution, and often times, a distribution will not have all 3.

Location (Loc)

The location parameter, often denoted as μ (mu), shifts the entire distribution along the x-axis. It represents the central tendency or "location" of the distribution, determining where the bulk of the data is centered.

Example

In the normal distribution, the location parameter is the mean, (\mu). If \(\mu=0\), the distribution is centered around 0; if \(\mu=5\), the distribution shifts to be centered around 5. The probability density function for the normal is given by: $$f(x;\mu,\sigma^2)=\frac{1}{\sqrt{2\pi\sigma^2}}\exp\left(-\frac{(x-\mu)^2}{2\sigma^2}\right)$$ The loc parameter can be thought of as a coordinate shift through substitution of variables.

Scale

The scale parameter controls the spread or dispersion of the distribution. It stretches or compresses the distribution horizontally, determining how tightly or loosely the values cluster around the center. In the above example of the normal distribution, \(\sigma\) is the scale parameter.

Shape

The shape parameter influences the form or structure of a distribution.

Measure Practice Problems