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PROBABILITY DISTRIBUTIONS OF REAL-VALUED RANDOM VARIABLES

Because a probability distribution Pr on the real line is determined by the probability of being in a half-open interval Pr(''a'', ''b''], the probability distribution of a real-valued random variable ''X'' is completely characterized by its Cumulative Distribution Function :

: F(x) = \Pr \left X \le x ight \qquad orall x \in \mathbb{R}.


Discrete probability distribution

See Also: Discrete probability distribution


A probability distribution is called ''discrete'' if its cumulative distribution function only increases in jumps.

The Set of all values that a discrete random variable can assume with non-zero probability is either Finite or Countably Infinite because the sum of uncountably many positive Real Number s (which is the smallest upper bound of the set of all finite partial sums) always diverges to infinity. Typically, the set of possible values is topologically discrete in the sense that all its points are Isolated Point s. But, there are discrete random variables for which this countable set is Dense on the real line.

Discrete distributions are characterized by a Probability Mass Function , p such that
:
F(x) = \Pr \left \le x ight = \sum_{x_i \le x} p(x_i).



Continuous probability distribution

See Also: Continuous probability distribution


By one convention, a probability distribution is called ''continuous'' if its cumulative distribution function is Continuous , which means that it belongs to a random variable ''X'' for which Pr ''X'' = ''x'' = 0 for all ''x'' in R.

Another convention reserves the term ''continuous probability distribution'' for function f defined on the real numbers such that

:
F(x) = \Pr \left X \le x ight = \int_{-\infty}^x f(t)\,dt


Discrete distributions and some continuous distributions (like the Devil's Staircase ) do not admit such a density.


TERMINOLOGY

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.

A discrete random variable is a random variable whose probability distribution is discrete. Similarly, a '''continuous random variable''' is a random variable whose probability distribution is continuous.


LIST OF IMPORTANT PROBABILITY DISTRIBUTIONS


Certain random variables occur very often in probability theory, in some cases due to their application to many natural and physical processes, and in some cases due to theoretical reasons such as the Central Limit Theorem , the Poisson Limit Theorem , or properties such as Memorylessness or other Characterizations . Their distributions therefore have gained ''special importance'' in probability theory.


Discrete distributions


With finite support


  • 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 because although its output is determinate, its input is random. 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'' 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 .



With infinite support






Continuous distributions


Supported on a bounded interval

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Supported on semi-infinite intervals, usually [0,∞)

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Supported on the whole real line

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Joint distributions


For any set of Independent random variables the Probability Density Function of their Joint Distribution is the product of their individual density functions.


Two or more random variables on the same sample space




Matrix-valued distributions




Miscellaneous distributions




DEMONSTRATIONS AND ACTIVITIES

The SOCR resource provides web-based tools (applets) for sampling from and interacting with many of these discrete and continuous distributions. Also, a number of distribution-specific activities are provided that demonstrate the utilization of general probability distributions.


SEE ALSO





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