- The Bernoulli Distribution , which takes value 1 with probability ''p'' and value 0 with probability ''q'' = 1 − ''p''.
- The Rademacher distribution, which takes value 1 with probability 1/2 and value −1 with probability 1/2.
- The Binomial Distribution , which describes the number of successes in a series of independent Yes/No experiments.
- The Degenerate Distribution at ''x''0, where ''X'' is certain to take the value ''x0''. This does not look random, but it satisfies the definition of Random Variable . This is useful because it puts deterministic variables and random variables in the same formalism.
- The Discrete Uniform Distribution , where all elements of a finite Set are equally likely. This is supposed to be the distribution of a balanced coin, an unbiased die, a casino roulette, or the first card of a well-shuffled deck. Also, one can use measurements of quantum states to generate uniform random variables. All these are "physical" or "mechanical" devices, subject to design flaws or perturbations, so the uniform distribution is only an approximation of their behaviour. In digital computers, Pseudo-random Number Generators are used to produced a Statistically Random discrete uniform distribution.
- The Hypergeometric Distribution , which describes the number of successes in the first ''m'' of a series of ''n'' independent Yes/No experiments, if the total number of successes is known.
- Zipf's Law or the Zipf distribution. A discrete Power-law distribution, the most famous example of which is the description of the frequency of words in the English language.
- The Zipf-Mandelbrot Law is a discrete power law distribution which is a generalization of the Zipf Distribution .
- Fisher's Noncentral Hypergeometric Distribution
- Wallenius' Noncentral Hypergeometric Distribution
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- The Beta Prime Distribution
- The Cauchy Distribution , an example of a distribution which does not have an Expected Value or a Variance . In physics it is usually called a Lorentzian Profile , and is associated with many processes, including Resonance energy distribution, impact and natural Spectral Line broadening and quadratic Stark line broadening.
- The Fisher-Tippett , extreme value, or log-Weibull distribution
- --- The Gumbel Distribution , a special case of the Fisher-Tippett distribution
- The Generalized Extreme Value Distribution
- The Hyperbolic Secant Distribution
- The Landau Distribution
- The Laplace Distribution
- The Lévy Skew Alpha-stable Distribution is often used to characterize financial data and critical behavior.
- The Map-Airy Distribution
- The Normal Distribution , also called the Gaussian or the bell curve. It is ubiquitous in nature and statistics due to the Central Limit Theorem : every variable that can be modelled as a sum of many small independent variables is approximately normal.
- The Skew-normal Distribution
- Student's T-distribution , useful for estimating unknown means of Gaussian populations.
- --- The Noncentral T-distribution
- The Type-1 Gumbel Distribution
- The Voigt Distribution , or Voigt profile, is the convolution of a Normal Distribution and a Cauchy Distribution . It is found in spectroscopy when Spectral Line profiles are broadened by a mixture of Lorentzian and Doppler broadening mechanisms.
For any set of Independent random variables the Probability Density Function of the joint distribution is the product of the individual ones.
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