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Home > Probability density function

Formally, a probability distribution has density f(x) if f(x) is a non-negative Lebesgue-integrable function R → R such that the probability of the interval [a, b] is given by

for any two numbers a and b. This implies that the total integral of f must be 1. Conversely, any non-negative Lebesgue-integrable function with total integral 1 is the probability density of a suitably defined probability distribution.

For example, the continuous uniform distribution on the interval [0,1] has probability density f(x) = 1 for 0 ≤ x ≤ 1 and zero elsewhere. The standard normal distribution has probability density

If a random variable X is given and its distribution admits a probability density function f(x), then the expected value of X (if it exists) can be calculated as

Not every probability distribution has a density function: the distributions of discrete random variables do not; nor does the Cantor distribution, even though it has no discrete component, i.e., does not assign positive probability to any individual point.

A distribution has a density function if and only if its cumulative distribution function F(x) is absolutely continuous. In this case, F is almost everywhereIn measure theory (a branch of mathematical analysis), one says that a property holds almost everywhere if the set of elements for which the property does not hold is a null set, i. is a set with measure zero. If used for properties of the real numbers, t differentiableCalculus In mathematics, the derivative of a function is one of the two central concepts of calculus. The inverse of a derivative is called the antiderivative, or indefinite integral. The derivative of a function at a certain point is a measure of the rat, and its derivative can be used as probability density:

If a probability distribution admits a density, then the probability of every one-point set {a} is zero.

It is a common mistake to think of f(a) as the probability of {a}, but this is incorrect; in fact, f(a) will often be bigger than 1 - consider a random variable with a uniform distribution between 0 and 1/2.

Two probability densities f and g represent the same probability distribution precisely if they differ only on a set of LebesgueMeasure theory The Lebesgue measure is the standard way of assigning a volume to subsets of Euclidean space. It is used throughout real analysis, in particular to define Lebesgue integration. Sets which can be assigned a volume are called Lebesgue measura measure zeroA set of numbers has measure zero if its Lebesgue measure is zero. A classic example is the set of rational numbers in the real line under standard (Euclidean) Lebesgue measure. As another example, a monotonic function on the real line is continuous every.

See also

• likelihoodIn statistics, a likelihood function is a conditional probability function considered a function of its second argument with its first argument held fixed, thus: : and also any other function proportional to such a function. That is, the likelihood functi
• probability mass functionIn probability theory, a probability mass function (abbreviated pmf gives the probability that a discrete random variable is exactly equal to some value. A probability mass function differs from a probability density function in that the values of the lat
• exponential familyIn statistics, the exponential family of probability density functions or probability mass functions comprises those that have the following form: : where: h ''x is the reference density η is the natural parameter a column vector, so that ηT, its

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ProbabilityThe word probability derives from the Latin probare (to prove, or to test). Informally, probable is one of several words applied to uncertain events or knowledge, being more or less interchangeable with likely risky hazardous uncertain and doubtful depend Probability interpretationsThe word probability has been used in a variety of ways since it was first coined in relation to games of chance. There are two broad categories of probability interpretations: Frequentists assign probabilities only to random events according to their rel Probability axiomsThe probability of some event (denoted ) is defined with respect to a "universe" or sample space of all possible elementary events in such a way that must satisfy the Kolmogorov axioms. Alternatively, a probability can be interpreted as a measure on a &si Probability theoryProbability theory Discrete mathematics Mathematical analysis Probability theory is the mathematical study of probability. Mathematicians think of probabilities as numbers in the interval from 0 to 1 assigned to "events" whose occurrence or failure to occ Probability distributionIn mathematics, a probability distribution assigns to every interval of the real numbers a probability, so that the probability axioms are satisfied. In technical terms, a probability distribution is a probability measure whose domain is the Borel algebra Probability spaceIn mathematics, a probability space is a set S together with a σ-algebra X on S and a measure P on that σ-algebra such that P ''S 1. The set S is called the sample space and the elements of X are called the events. The measure P is called the Probability density function Probability and statistics Probability mass function Probability vector Probability-generating function Probability amplitude Probability of error Probability of kill