In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

 and statistics

Statistics

Statistics is the study of the collection, organization, analysis, and interpretation of data. It deals with all aspects of this, including the planning of data collection in terms of the design of surveys and experiments....

, the U-quadratic distribution is a continuous probability distribution

Probability distribution

In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....

 defined by a unique quadratic function with lower limit a and upper limit b.

Parameter relations

This distribution has effectively only two parameters a, b, as the other two are explicit functions of the support defined by the former two parameters:

(gravitational balance center, offset), and

(vertical scale).

Related distributions

One can introduce a vertically inverted ()-quadratic distribution in analogous fashion.

Applications

This distribution is a useful model for symmetric bimodal processes. Other continuous distributions allow more flexibility, in terms of relaxing the symmetry and the quadratic shape of the density function, which are enforced in the U-quadratic distribution - e.g., Beta distribution, Gamma distribution, etc.

Moment generating function

Characteristic function