Continuous probability distribution

In probability theory, a probability distribution is called continuous if its cumulative distribution function is continuous. That is equivalent to saying that for random variables "X" with the distribution in question, Pr ["X" = "a"] = 0 for all real numbers "a", i.e.: the probability that "X" attains the value "a" is zero, for any number "a". If the distribution of "X" is continuous then "X" is called a continuous random variable.

While for a discrete probability distribution one could say that an event with probability zero is impossible (e.g. throwing the dice and getting e.g. 3.5: this has probability zero, and (or because) it is impossible), this cannot be said in the case of a continuous random variable, because then no value would be possible (e.g. to measure the width of an oak leaf, and get e.g. 3.5 cm: this is possible, but the 'exact' value of 3.5cm has probability zero, because there are infinitely many exact values even between 3cm and 4cm. Every single one of these exact values has probability zero. Only an interval, eg. the one between 3cm and 4cm may have a probability greater than zero.). This paradox is resolved by realizing that the probability that "X" attains some value within an uncountable set (for example an interval) cannot be found by adding the probabilities for individual values. Intuitively you could say, every single exact value has an infinitesimally small probability, but strictly spoken that's what we call zero.

Under an alternative and stronger definition, the term "continuous probability distribution" is reserved for distributions that have probability density functions. These are most precisely called absolutely continuous random variables (see Radon–Nikodym theorem). For a random variable X, being absolutely continuous is equivalent to saying that the probability that X attains a value in any given subset S of its range with Lebesgue measure zero is equal to zero. This does not follow from the condition Pr ["X" = "a"] = 0 for all real numbers "a", since there are uncountable sets with Lebesgue-measure zero (e.g. the Cantor set).

A random variable with the Cantor distribution is continuous according to the first convention, but according to the second, it is not (absolutely) continuous. Also, it is not discrete nor a weighted average of discrete and absolutely continuous random variables.

In practical applications, random variables are often either discrete or absolutely continuous, although mixtures of the two also arise naturally.

The normal distribution, continuous uniform distribution, Beta distribution, and Gamma distribution are well known absolutely continuous distributions. The normal distribution, also called the Gaussian or the bell curve, 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.

References

External links

* http://webpages.dcu.ie/~applebyj/ms207/CNSRV1.pdf


Wikimedia Foundation. 2010.

Look at other dictionaries:

  • Probability distribution — This article is about probability distribution. For generalized functions in mathematical analysis, see Distribution (mathematics). For other uses, see Distribution (disambiguation). In probability theory, a probability mass, probability density …   Wikipedia

  • Joint probability distribution — In the study of probability, given two random variables X and Y that are defined on the same probability space, the joint distribution for X and Y defines the probability of events defined in terms of both X and Y. In the case of only two random… …   Wikipedia

  • Conditional probability distribution — Given two jointly distributed random variables X and Y, the conditional probability distribution of Y given X is the probability distribution of Y when X is known to be a particular value. If the conditional distribution of Y given X is a… …   Wikipedia

  • normal probability distribution — A probability distribution for a continuous random variable that forms a symmetrical bell shaped curve around the mean. This distribution has no skewness or excess kurtosis. Bloomberg Financial Dictionary …   Financial and business terms

  • Normal probability distribution — A probability distribution for a continuous random variable that is forms a symmetrical bell shaped curve around the mean. The New York Times Financial Glossary …   Financial and business terms

  • Maximum entropy probability distribution — In statistics and information theory, a maximum entropy probability distribution is a probability distribution whose entropy is at least as great as that of all other members of a specified class of distributions. According to the principle of… …   Wikipedia

  • Probability theory — is the branch of mathematics concerned with analysis of random phenomena.[1] The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non deterministic events or measured… …   Wikipedia

  • Probability density function — Boxplot and probability density function of a normal distribution N(0, σ2). In probability theory, a probability density function (pdf), or density of a continuous random variable is a function that describes the relative likelihood for this… …   Wikipedia

  • probability theory — Math., Statistics. the theory of analyzing and making statements concerning the probability of the occurrence of uncertain events. Cf. probability (def. 4). [1830 40] * * * Branch of mathematics that deals with analysis of random events.… …   Universalium

  • Distribution (mathematics) — This article is about generalized functions in mathematical analysis. For the probability meaning, see Probability distribution. For other uses, see Distribution (disambiguation). In mathematical analysis, distributions (or generalized functions) …   Wikipedia

Share the article and excerpts

Direct link
Do a right-click on the link above
and select “Copy Link”