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  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” .
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.
 K. Pearson. (1892). The Grammar of Science. (Third edition available here).
 Ramchandran, K. & Tsokos, C. Mathematical Statistics and Applications.
 Walli, G. (2010). Bayesian Variable Selection in Normal Regression Models. Thesis.