- Continuous probability distribution
probability theory, a probability distributionis called continuous if its cumulative distribution functionis 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 distributionone could say that an event with probabilityzero 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 paradoxis resolved by realizing that the probability that "X" attains some value within an uncountableset (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 continuousrandom 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 subsetS of its range with Lebesgue measurezero 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 distributionis 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.
normal distribution, continuous uniform distribution, Beta distribution, and Gamma distributionare 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.
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