Probability density function

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 random variable to take on a given value. The probability of the random variable falling within a particular range of values is given by the integral of this variable’s density over that range—that is, it is given by the area under the density function but above the horizontal axis and between the lowest and greatest values of the range. The probability density function is nonnegative everywhere, and its integral over the entire space is equal to one.

The terms "probability distribution function" and "probability function" have also sometimes been used to denote the probability density function. However, this use is not standard among probabilists and statisticians. In other sources, "probability distribution function" may be used when the probability distribution is defined as a function over general sets of values, or it may refer to the cumulative distribution function, or it may be a probability mass function (PMF) rather than the density. Further confusion of terminology exists because density function has also been used for what is here called the "probability mass function" (PMF). In general though, the PMF is used in the context of discrete random variables (random variables that take values on a discrete set), while PDF is used in the context of continuous random variables.

Radon–Nikodym theorem

In mathematics, the Radon–Nikodym theorem is a result in measure theory which states that, given a measurable space , if a σ-finite measure on is absolutely continuous with respect to a σ-finite measure μ on , then there is a measurable function , such that for any measurable subset ,

The function  f  is called the Radon–Nikodym derivative and denoted by .

The theorem is named after Johann Radon, who proved the theorem for the special case where the underlying space is R^N in 1913, and for Otto Nikodym who proved the general case in 1930. In 1936 Hans Freudenthal further generalized the Radon–Nikodym theorem by proving the Freudenthal spectral theorem, a result in Riesz space theory, which contains the Radon–Nikodym theorem as a special case.

If Y is a Banach space and the generalization of the Radon–Nikodym theorem also holds for functions with values in Y (mutatis mutandis), then Y is said to have the Radon–Nikodym property. All Hilbert spaces have the Radon–Nikodym property.