Density of a probability distribution

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probability density

The derivative of the distribution function corresponding to an absolutely-continuous probability measure.

Let be a random vector taking values in an -dimensional Euclidean space , let be its distribution function, and let there exist a non-negative function such that

for any real . Then is called the probability density of , and for any Borel set ,

Any non-negative integrable function satisfy the condition

is the probability density of some random vector.

If two random vectors and taking values in are independent and have probability densities and respectively, then the random vector has the probability density that is the convolution of and :

Let and be random vectors taking values in and and having probability densities and respectively, and let be a random vector in . If then and are independent, has the probability density , which is called the joint probability density of the random vectors and , where


Conversely, if has a probability density that satisfies (1), then and are independent.

The characteristic function of a random vector having a probability density is expressed by

where if is absolutely integrable then is a bounded continuous function, and

The probability density and the corresponding characteristic function are related also by the following relation (Plancherel's identity): The function is integrable if and only if the function is integrable, and in that case

Let be a measurable space, and let and be -finite measures on with absolutely continuous with respect to , i.e. implies , . In that case there exists on a non-negative measurable function such that

for any . The function is called the Radon–Nikodým derivative of with respect to , while if is a probability measure, it is also the probability density of relative to .

A concept closely related to the probability density is that of a dominated family of distributions. A family of probability distributions on a measurable space is called dominated if there exists a -finite measure on such that each probability measure from has a probability density relative to (or, what is the same, if each measure from is absolutely continuous with respect to ). The assumption of dominance is important in certain theorems in mathematical statistics.


[1] Yu.V. [Yu.V. Prokhorov] Prohorov, Yu.A. Rozanov, "Probability theory, basic concepts. Limit theorems, random processes" , Springer (1969) (Translated from Russian)
[2] W. Feller, "An introduction to probability theory and its applications" , 2 , Wiley (1971)
[3] E.L. Lehmann, "Testing statistical hypotheses" , Wiley (1986)
How to Cite This Entry:
Density of a probability distribution. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by N.G. Ushakov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article