A probability distribution is a special case of the more general notion of a Probability Measure , which is a function that assigns probabilities satisfying the Kolmogorov Axioms to the measurable sets of a Measurable Space . Additionally, some authors define a distribution generally as the probability measure induced by a Random Variable ''X'' on its Range - the probability of a set ''B'' is . However, this article discusses only probability measures over the real numbers.
Every random variable gives rise to a probability distribution, and this distribution contains most of the important information about the variable. If ''X'' is a random variable, the corresponding probability distribution assigns to the interval ''b'' the probability Pr ≤ ''X'' ≤ ''b'' , i.e. the probability that the variable ''X'' will take a value in the interval ''b'' .
The probability distribution of the variable ''X'' can be uniquely described by its Cumulative Distribution Function ''F''(''x''), which is defined by
:
for any ''x'' in .
A distribution is called ''discrete'' if its cumulative distribution function consists of a sequence of finite jumps, which means that it belongs to a function ''f'' defined on the real numbers such that
:
for all ''a'' and ''b''. Of course, discrete distributions do not admit such a density; there also exist some continuous distributions like the Devil's Staircase that do not admit a density.
- The ''support'' of a distribution is the smallest closed set whose complement has probability zero.
- The probability distribution of the sum of two independent random variables is the Convolution of each of their distributions.
- The probability distribution of the difference of two random variables is the Cross-correlation of each of their distributions.
Several probability distributions are so important in theory or applications that they have been given specific names:
- 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 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 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 produce 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 .
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- Fisher's Z-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.
- 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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