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.

Parameter Families for Common Distributions
Distribution	Location ($\mu$)	Scale ($\sigma$)	Shape ($\xi$, $\alpha$, etc.)
Normal	$\mu$	$\sigma$	—
Exponential	—	$\lambda^{-1}$	—
Gamma	—	$\theta$	$\alpha$
Weibull	—	$\lambda$	$\kappa$
Gumbel	$\mu$	$\beta$	—
Fréchet	—	$\sigma$	$\alpha$
Cauchy	$x_0$	$\gamma$	—
Beta	—	—	$\alpha, \beta$
Pareto	$x_m$	—	$\alpha$
Log-normal	$\mu$	$\sigma$	—
Student’s t	$\mu$	$\sigma$	$\nu$
Generalized Pareto	$\mu$	$\sigma$	$\xi$
Laplace	$\mu$	$b$	—
Rayleigh	—	$\sigma$	—
Beta prime	—	—	$\alpha, \beta$
Burr	—	$c$	$k$
Logistic	$\mu$	$s$	—

Interactive Probability Distribution Plotter

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Parameter Family Practice Problems

Look up a distribution not listed in the table above and try to classify its parameters.
Describe the difference between the location, scale, and shape parameter types.
For a normal distribution
Effect of Location and Scale Parameters:
1. Consider a normal distribution $X \sim N(\mu, \sigma^2)$ with location parameter $\mu$ and scale parameter $\sigma$.
  1. How does changing the value of $\mu$ affect the shape or spread of the distribution?
  2. How does changing the value of $\sigma$ affect the shape or location of the distribution?
  3. Plot the probability density function (PDF) of the normal distribution for different values of $\mu$ and $\sigma$, and describe the changes in the graph.
Parameter Estimation:
1. Given the following dataset, assume that it comes from a Laplace distribution. Estimate the location and scale parameters.
  $\{-0.5, 1.3, -1.2, 0.7, -0.3, 1.0, -1.5, 0.6\}$
  Hint: The location parameter $\mu$ is the median of the dataset, and the scale parameter $b$ is the average absolute deviation from $\mu$.
Transformations of Random Variables:
1. Let $X \sim \text{Weibull}(\lambda, k)$ where $\lambda$ is the scale parameter and $k$ is the shape parameter.
  1. Define a new random variable $Y = aX + b$, where $a > 0$ and $b$ are constants. What are the location, scale, and shape parameters of $Y$?
  2. Prove that the shape parameter remains unchanged, while the location and scale parameters are affected by the transformation.
Effect of Shape Parameters:
1. The Beta distribution has two shape parameters, $\alpha$ and $\beta$.
  1. Investigate how the distribution changes when $\alpha = \beta = 1$, $\alpha > \beta$, and $\alpha < \beta$. Plot the PDF of the Beta distribution for several combinations of $\alpha$ and $\beta$.
  2. Explain how the shape parameters control the skewness of the distribution.
Parameter Estimation for Scale and Shape:
1. Consider the Rayleigh distribution with scale parameter $\sigma$. Given the dataset:
  $\{2.5, 3.1, 2.8, 3.5, 2.9\}$
  Estimate the value of $\sigma$ using the maximum likelihood estimation (MLE) method.
Parameter Relations Across Distributions:
1. Some distributions can be expressed in terms of others. For example, the normal distribution and the log-normal distribution are closely related. Given that $Y \sim \text{LogNormal}(\mu, \sigma)$, derive the location and scale parameters of $X = \log(Y)$, where $X \sim N(\mu, \sigma^2)$.
  1. Explain how the location and scale parameters of $X$ relate to those of $Y$.
Impact of Parameters on Moments:
1. Consider the Burr distribution with parameters $c$ and $k$. Derive expressions for the mean and variance of the Burr distribution in terms of $c$ and $k$. Explain how changes in the parameters affect the mean and variance of the distribution.

Distribution Textbook (Work in Progress)

by John Della Rosa