< List of probability distributions

The **location scale family of distributions** incorporates popular distributions like the normal and uniform distribution, as well as several other distributions with special cases that belong to the family. In this article, we’ll dive deep into understanding the location scale family of distributions, its properties, and its significance in statistical analysis.

## What is the location scale family?

The location scale family of distributions refers to a group of probability distributions that can be transformed into each other through simple linear transformations. This means that if we apply a shift and scaling transformation to a given distribution, we can convert it into another distribution within the same family. For example, if we take a normal distribution and subtract its mean and divide it by its standard deviation, we obtain a standard normal distribution.

A random variable belongs to the location-scale family if its cumulative distribution function (CDF), F_{X} (x | a, b):= Pr(X ≤ | a, b) is a function only of (x – a)/b [1]:

Where:

*F(·)*is a distribution without any other parameters,*a*is the location parameter,*b*is the scale parameter.

If *b* = 1, the family is a location family — a subfamily of the location-scale family. If *a *= 0, the subfamily is a scale family with parameter *b*.

## Family members

Some other popular distributions within the location-scale family include the uniform distribution, the exponential distribution, the normal distribution and the Cauchy distribution. We can describe each of these distributions by adjusting two parameters: the location parameter and the scale parameter. The location parameter determines the position of the distribution on the number line, while the scale parameter determines the shape and dispersion of the distribution.

Other family members include

- Arcsine distribution,
- Cosine distribution,
- Extreme value distributions,
- Half-Cauchy distribution,
- Half-logistic distribution,
- Half-normal distribution,
- Hyperbolic secant distribution,
- Laplace distribution,
- Logistic distribution,
- Maxwell-Boltzmann distribution,
- Parabolic distributions of order 2,
- Rayleigh distribution,
- Semi–elliptical distribution,
- Teissier distribution,
- Triangular Distributions
- U–shaped and inverted U–shaped distributions,
- V–shaped distributions.

One important application of the location-scale family of distributions is in statistical modeling. By fitting data to a specific distribution within the family, we can make probabilistic predictions and conduct hypothesis testing. For example, if we have a set of data that we believe follows a normal distribution, we can estimate the parameters of the distribution and use it for making predictions about future observations. Similarly, if we have a set of data that follows a uniform distribution, we know that every value within a certain range is equally likely to occur.

## Special Cases and related distributions

The location-scale family of distributions also includes special cases of distributions that have their own unique properties. For example, the beta distribution is not in the scale-location family because it has two “extra” parameters: c and d. However, if we set both of these parameters to 1, the distribution becomes a uniform distribution, which is a “true” member of the location-scale family. Similarly, the gamma distribution has three parameters, but we can convert it to a chi-distribution by setting one of the parameters to 2.

Other distributions can be transformed or otherwise tweaked to join the family. For example, an exponential distribution that is reflected over the *x*-axis at *x *= *a *will become a location-scale distribution.

A related family is the log-location scale family, which includes the lognormal distribution, the loglogistic distribution, and the Weibull distribution. A distribution is in the log-location scale family if Y = log(T) is a member of the location-scale family [2].

In conclusion, the location-scale family of distributions is a powerful tool for understanding probability distributions and their relationships to each other. By adjusting two parameters, we can describe a wide range of distributions including the normal and uniform distributions. This family of distributions is also useful for statistical modeling and hypothesis testing. It’s essential for college students to have a solid understanding of the location-scale family of distributions in order to gain the skills needed for jobs and research in related fields.

## References

[1] The family of location–scale distributions. Retrieved December 15, 2021 from: http://geb.uni-giessen.de/geb/volltexte/2010/7607/pdf/RinneHorst_LocationScale_2010.pdf

[2] Location-Scale-Base Parametric Distributions.