Probability density

Probability density is one of the ways of defining a probability measure on a Euclidean space. . In the case when the probability measure is the distribution of a random variable, we speak about the density of the random variable .

Probability Density [edit]

Let be is a probability measure on , that is, probabilistic space is defined where denotes the Borel σ-algebra on . Let be denotes the Lebesgue measure on .

Definition 1. Probability called absolutely continuous (relative to Lebesgue measure) ( ), if any Borel set of zero Lebesgue measure also has a zero probability:

If the probability is absolutely continuous, then, according to the Radon – Nikodym theorem, there exists a nonnegative Borel function such that

,

where common abbreviation is used and the integral is understood in the sense of Lebesgue.

Definition 2. More generally, let Is an arbitrary measurable space, and and - two measures in this space. If there is a non-negative allowing to express measure through measure as

then this function is called the measure density as , or a derivative of Radon-Nikodym measures regarding measure , and denote

.

Probability Density Properties [edit]

• Probability density is defined almost everywhere. If a is probability density and almost everywhere with respect to Lebesgue measure, then the function also is the probability density .

• The integral of the density over the entire space is equal to unity:

.

Back if - non-negative i.e. function such that then there is an absolutely continuous probability measure on such that is its density.

• Substitution of a measure in the Lebesgue integral:

,

Where any Borel function that is integrable with respect to a probability measure .

Random Density [edit]

Let an arbitrary probability space be defined. and random variable (or random vector). induces a probability measure on called the random distribution .

Definition 3. If the distribution absolutely continuous with respect to Lebesgue measure, then its density called random density . The random variable itself called absolutely continuous.

Thus for an absolutely continuous random variable we have:

.

Remarks [edit]

• Not every random variable is absolutely continuous. Any discrete distribution, for example, is not absolutely continuous with respect to Lebesgue measure, and therefore discrete random variables have no density.

• Distribution function of an absolutely continuous random variable continuous and can be expressed in terms of density as follows:

.

In the one-dimensional case:

.

If a then and

.

In the one-dimensional case:

.

• The expectation of a function from an absolutely continuous random variable can be written in the form:

,

Where - Borel function, so defined and of course.

Conversion density of a random variable [edit]

Let be - an absolutely continuous random variable, and - injective continuously differentiable function such that where - Jacobian functions at the point . Then a random variable also absolutely continuous, and its density is:

.

In the one-dimensional case:

.

Examples of absolutely continuous distributions [edit]

• Beta distribution;
• Weibull distribution;
• Gamma distribution;
• Cauchy distribution;
• Lognormal distribution;
• Normal distribution;
• Continuous uniform distribution
• Pareto distribution;
• Student's distribution;
• Fisher distribution;
• Chi-square distribution;
• Exponential distribution;
• Multidimensional normal distribution.