A continuous probability distribution is a probability distribution that has a probability density function. Mathematicians also call such a distribution absolutely continuous, since its cumulative distribution function is absolutely continuous with respect to the Lebesgue measure λ. If the distribution of X is continuous, then X is called a continuous random variable. There are many examples of continuous probability distributions: normal, uniform, chi-squared, and others.
Intuitively, a continuous random variable is the one which can take a continuous range of values — as opposed to a discrete distribution, where the set of possible values for the random variable is at most countable. While for a discrete distribution an event with probability zero is impossible (e.g. rolling 3½ on a standard die is impossible, and has probability zero), this is not so in the case of a continuous random variable. For example, if one measures the width of an oak leaf, the result of 3½ cm is possible, however it has probability zero because there are uncountably many other potential values even between 3 cm and 4 cm. Each of these individual outcomes has probability zero, yet the probability that the outcome will fall into the interval (3 cm, 4 cm) is nonzero. This apparent paradox is resolved by the fact that the probability that X attains some value within an infinite set, such as an interval, cannot be found by naively adding the probabilities for individual values. Formally, each value has an infinitesimally small probability, which statistically is equivalent to zero.
Formally, if X is a continuous random variable, then it has a probability density function ƒ(x), and therefore its probability of falling into a given interval, say is given by the integral
In particular, the probability for X to take any single value a (that is a ≤ X ≤ a) is zero, because an integral with coinciding upper and lower limits is always equal to zero.
The definition states that a continuous probability distribution must possess a density, or equivalently, its cumulative distribution function be absolutely continuous. This requirement is stronger than simple continuity of the cdf, and there is a special class of distributions, singular distributions, which are neither continuous nor discrete nor their mixture. An example is given by the Cantor distribution. Such singular distributions however are never encountered in practice.
Note on terminology: some authors use the term "continuous distribution" to denote the distribution with continuous cdf. Thus, their definition includes both the (absolutely) continuous and singular distributions.
By one convention, a probability distribution is called continuous if its cumulative distribution function is continuous and, therefore, the probability measure of singletons for all .
Another convention reserves the term continuous probability distribution for absolutely continuous distributions. These distributions can be characterized by a probability density function: a non-negative Lebesgue integrable function defined on the real numbers such that
![F(x) = \mu(-\infty,x] = \int_{-\infty}^x f(t)\,dt.](http://upload.wikimedia.org/math/4/1/8/41882dedea3ba5430b8267370c175c4d.png)
Discrete distributions and some continuous distributions (like the Cantor distribution) do not admit such a density.
Read more about this topic: Probability Distribution
Famous quotes containing the words continuous, probability and/or distribution:
“For good and evil, man is a free creative spirit. This produces the very queer world we live in, a world in continuous creation and therefore continuous change and insecurity.”
—Joyce Cary (1888–1957)
“Crushed to earth and rising again is an author’s gymnastic. Once he fails to struggle to his feet and grab his pen, he will contemplate a fact he should never permit himself to face: that in all probability books have been written, are being written, will be written, better than anything he has done, is doing, or will do.”
—Fannie Hurst (1889–1968)
“The question for the country now is how to secure a more equal distribution of property among the people. There can be no republican institutions with vast masses of property permanently in a few hands, and large masses of voters without property.... Let no man get by inheritance, or by will, more than will produce at four per cent interest an income ... of fifteen thousand dollars] per year, or an estate of five hundred thousand dollars.”
—Rutherford Birchard Hayes (1822–1893)