Systems of probability distributions

A system of probability distributions refers to a collection or framework of probability distributions that describe the probabilities of different outcomes in a certain context, often involving multiple random variables or scenarios. This concept can be applied in various fields such as statistics, machine learning, economics, and decision theory. Here are several key aspects related to systems of probability distributions: 1. **Joint Distributions**: This refers to the probability distribution that covers multiple random variables simultaneously.

The Burr distribution, also known as the Burr Type XII distribution, is a probability distribution that is used in statistics to model a variety of phenomena. It is characterized by its flexibility, allowing it to fit a wide range of data types. The Burr distribution is defined by its cumulative distribution function (CDF) and can be parameterized in several ways, generally using two shape parameters (often denoted as \(k\) and \(c\)).

The Metalog distribution is a flexible family of probability distributions that can be used to model various types of data. It was introduced by T. H. D. U. Chen et al. in a 2012 paper as a way to provide a more versatile alternative to traditional distributions like the normal, lognormal, or gamma distributions.

A mixture distribution is a probabilistic model that represents a distribution as a combination of two or more component distributions, each of which is weighted by a certain probability. This approach is useful in various fields, including statistics, machine learning, and data analysis, as it allows for modeling complex data patterns that cannot be easily captured by a single distribution. ### Key Characteristics: 1. **Components**: Each component of the mixture can be a different distribution (e.g.

The Pearson distribution, or Pearson system of distributions, is a family of continuous probability distributions that are defined based on moments, especially how the shape of the distribution is determined by its moments (mean, variance, skewness, and kurtosis). This system was introduced by Karl Pearson in the early 20th century, and it encompasses a wide range of probability distributions, including the normal distribution, beta distribution, and skewed distributions.

A **quantile-parameterized distribution** is a type of probability distribution that is characterized directly in terms of its quantiles, rather than through its probability density function (PDF) or cumulative distribution function (CDF). This approach emphasizes the distribution's quantile function, which provides a way to describe the distribution based on the values at specified probabilities.

The Tweedie distribution is a family of probability distributions that generalizes several well-known distributions, including the normal, Poisson, gamma, and inverse Gaussian distributions. It is characterized by a parameter \(\p\) (the power parameter), which determines the specific type of distribution within the Tweedie family.