< List of probability distributions

The term “special distribution” refers to any probability distribution used frequently in practice. These distributions have special names and possess importance for their ability to model a range of real-life phenomena. They might also be associated with other special distributions through *conditioning *(an event treated as having occurred), limits (from calculus), or transformations. Consequently, these distributions are special because they are deemed “useful” or “important”. They commonly:

- Possess a simple form probability density functions (PDF) or probability mass function (PMF),
- Serve a wide range of applications and uses,
- Hold recognition within the fields they apply.

## History of the special distribution

In 1892, Karl Pearson introduced the phrase “special distribution” in his book *The Grammar of Science* [1] to describe a probability distribution used to model specific phenomena. For instance, the normal distribution is a popular choice to model natural phenomena like height, weight, and IQ scores, owing to its symmetric and bell-shaped curve that centralizes most data around the mean. Pearson referred to certain probability distributions as more suitable to model phenomena than others, based on the idea of appropriateness. As a side note, Pearson also coined the term “standard deviation” [2].

## Special distribution list

- Arcsine distribution: probability of a random variable taking on a value between 0 and 1.
- Benford’s distribution: probability of the first digit of a number in a dataset.
- Bernoulli Distribution: probability of a binary event occurring, such as a yes/no question.
- Beta distribution: probability of a random variable taking on a value between 0 and 1.
- Beta prime distribution: probability of a random variable taking on a value greater than 0.
- Binomial Distribution: number of successes in a sequence of
*n*independent experiments with a yes–no question, and each with its own Boolean-valued outcome: success or failure. - Cauchy distribution: often used to model data that is heavy-tailed.
- Chi-square distribution: describes the sum of squares of independent standard normal random variables
- Uniform distribution: a flat topped describes the probability of a random variable taking on any value within a specified range.
- Erlang Distribution: time between events in a Poisson process.
- Exponential distributions: often used to model the time between phone calls, website requests, or other events that occur randomly in time.
- Exponential-logarithmic distribution: a continuous probability distribution that describes time to failure of a system subject to increasing and decreasing failure rates
- Extreme Value Distribution: describes the probability of extreme values, such as the maximum or minimum of a set of data.
- F-distribution: a continuous probability distribution that describes the ratio of two independent chi-square random variables.
- Fisk (Log-logistic) distribution.
- Folded normal distribution: created by taking the absolute value of a normally distributed random variable.
- Gamma distribution: a two-parameter continuous probability distribution that describes the time it takes for a random variable to reach a certain value.
- Geometric Distribution: describes the number of trials required to achieve the first success in a sequence of independent Bernoulli trials, with a constant probability of success for each trial.
- Gompertz distribution: describes the time it takes for a random variable to reach a certain value; has an exponentially increasing failure rate.
- Hyperbolic secant distribution: a continuous probability distribution with PDF and characteristic function that are proportional to the hyperbolic secant function.
- Irwin-Hall distribution: describes the sum of n independent and identically distributed random variables.
- Laplace distribution: a symmetric and continuous probability distribution with heavier tails and a sharper peak than the normal distribution.
- Lévy distribution: a continuous probability distribution that has heavy tails and no finite mean or variance.
- Location scale distribution: a family of probability distributions that can be generated by shifting and rescaling a single distribution.
- Logarithmic distribution: a long-tailed distribution that describes the number of times an event occurs in a fixed period of time.
- Logistic distribution: similar to the normal distribution, but has heavier tails.
- Log-normal distribution: a special distribution used to model values that are positive; has a long right tail.
- Maxwell-Boltzmann distribution: describes the speeds of particles in a gas.
- Normal distribution: a probability distribution that is symmetric about the mean. The “bell curve.”
- Pareto distribution: a right-skewed, long tail distribution often used to model the distribution of wealth, income, and other variables.
- Power series distribution: a discrete probability distribution constructed from a power series.
- Rayleigh distribution: used to model the magnitudes of random variables.
- Semicircle Distribution: often used to model the distribution of eigenvalues of random matrices.
- Sine distribution: a continuous probability distribution that is based on a portion of the sine curve.
- Stable distribution: a continuous probability distribution that is invariant under addition and scaling.
- Student’s t-distribution: a special distribution used to model the distribution of the mean of a sample when the population standard deviation is unknown.
- Triangular (Triangle) distribution: a continuous probability distribution shaped like a triangle
- U-power distribution: a continuous probability distribution that is symmetric about 1 with a power-law tail.
- Wald distribution: used to model the distribution of the ratio of two independent standard normal variables.
- Weibull distribution: used to model the distribution of time to failure or time between events.
- Zeta (Zipf) distribution: used to model the distribution of the size or ranks of certain types of objects randomly chosen from certain types of populations.

## References

[1] K. Pearson. (1892). The Grammar of Science. (Third edition available here).

[2] Ramchandran, K. & Tsokos, C. Mathematical Statistics and Applications.

[3] Walli, G. (2010). Bayesian Variable Selection in Normal Regression Models. Thesis.