probability theory Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set o ...

and

statistics Statistics (from German language, German: ''wikt:Statistik#German, Statistik'', "description of a State (polity), state, a country") is the discipline that concerns the collection, organization, analysis, interpretation, and presentation of ...

, the trapezoidal distribution is a continuous

probability distribution In probability theory and statistics, a probability distribution is the mathematical function that gives the probabilities of occurrence of different possible outcomes for an experiment. It is a mathematical description of a random phenomenon i ...

whose

probability density function In probability theory, a probability density function (PDF), or density of a continuous random variable, is a function whose value at any given sample (or point) in the sample space (the set of possible values taken by the random variable) can ...

graph resembles a

trapezoid A quadrilateral with at least one pair of parallel sides is called a trapezoid () in American and Canadian English. In British and other forms of English, it is called a trapezium (). A trapezoid is necessarily a Convex polygon, convex quadri ...

. Likewise, trapezoidal distributions also roughly resemble

mesa A mesa is an isolated, flat-topped elevation, ridge or hill, which is bounded from all sides by steep escarpments and stands distinctly above a surrounding plain. Mesas characteristically consist of flat-lying soft sedimentary rocks capped by ...

s or

plateau In geology and physical geography, a plateau (; ; ), also called a high plain or a tableland, is an area of a highland consisting of flat terrain that is raised sharply above the surrounding area on at least one side. Often one or more sides ha ...

s. Each trapezoidal distribution has a

lower bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element ...

and an

upper bound In mathematics, particularly in order theory, an upper bound or majorant of a subset of some preordered set is an element of that is greater than or equal to every element of . Dually, a lower bound or minorant of is defined to be an element ...

, where , beyond which no

values In ethics and social sciences, value denotes the degree of importance of something or action, with the aim of determining which actions are best to do or what way is best to live (normative ethics in ethics), or to describe the significance of dif ...

events Event may refer to: Gatherings of people * Ceremony, an event of ritual significance, performed on a special occasion * Convention (meeting), a gathering of individuals engaged in some common interest * Event management, the organization of eve ...

on the distribution can occur (i.e. beyond which the

probability Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...

is always zero). In addition, there are two sharp bending points (non-

differentiable In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non-vertical tangent line at each interior point in its ...

discontinuities) within the probability distribution, which we will call and , which occur between and , such that . The image to the right shows a perfectly

linear Linearity is the property of a mathematical relationship (''function'') that can be graphically represented as a straight line. Linearity is closely related to '' proportionality''. Examples in physics include rectilinear motion, the linear r ...

trapezoidal distribution. However, not all trapezoidal distributions are so precisely shaped. In the standard case, where the middle part of the trapezoid is completely flat, and the side ramps are perfectly linear, all of the values between and will occur with equal frequency, and therefore all such points will be

modes Mode ( la, modus meaning "manner, tune, measure, due measure, rhythm, melody") may refer to: Arts and entertainment * '' MO''D''E (magazine)'', a defunct U.S. women's fashion magazine * ''Mode'' magazine, a fictional fashion magazine which is ...

(local frequency maxima) of the distribution. On the other hand, though, if the middle part of the trapezoid is not completely flat, or if one or both of the side ramps are not perfectly linear, then the trapezoidal distribution in question is a generalized trapezoidal distribution, and more complicated and context-dependent rules may apply. The side ramps of a trapezoidal distribution are not required to be

symmetric Symmetry (from grc, συμμετρία "agreement in dimensions, due proportion, arrangement") in everyday language refers to a sense of harmonious and beautiful proportion and balance. In mathematics, "symmetry" has a more precise definiti ...

in the general case, just as the sides of trapezoids in

geometry Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is c ...

are not required to be symmetric. The non-central moments of the trapezoidal distribution are

= \frac\frac\left(\frac - \frac\right)

Special case In logic, especially as applied in mathematics, concept is a special case or specialization of concept precisely if every instance of is also an instance of but not vice versa, or equivalently, if is a generalization of . A limiting case is ...

s of the trapezoidal distribution include the uniform distribution (with and ) and the

triangular distribution In probability theory and statistics, the triangular distribution is a continuous probability distribution with lower limit ''a'', upper limit ''b'' and mode ''c'', where ''a'' < ''b'' and ''a'' ≤ ''c'' ≤ ''b''. ...

(with ). Trapezoidal probability distributions seem to not be discussed very often in the literature. The

uniform A uniform is a variety of clothing worn by members of an organization while participating in that organization's activity. Modern uniforms are most often worn by armed forces and paramilitary organizations such as police, emergency services, se ...

triangular A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry. A triangle with vertices ''A'', ''B'', and ''C'' is denoted \triangle ABC. In Euclidean geometry, any three points, when non- collinea ...

, Irwin-Hall,

Bates Bates may refer to: Places * Bates, Arkansas, an unincorporated community * Bates, Illinois. an unincorporated community in Sangamon County * Bates, Michigan, a community in Grand Traverse County * Bates, New York, a hamlet in the town of Elli ...

, Poisson,

normal Normal(s) or The Normal(s) may refer to: Film and television * ''Normal'' (2003 film), starring Jessica Lange and Tom Wilkinson * ''Normal'' (2007 film), starring Carrie-Anne Moss, Kevin Zegers, Callum Keith Rennie, and Andrew Airlie * ''Norma ...

, bimodal, and

multimodal distribution In statistics, a multimodal distribution is a probability distribution with more than one mode. These appear as distinct peaks (local maxima) in the probability density function, as shown in Figures 1 and 2. Categorical, continuous, and dis ...

s are all more frequently discussed in the literature. This may be because these other (non-trapezoidal) distributions seem to occur more frequently in nature than the trapezoidal distribution does. The normal distribution in particular is especially common in nature, just as one would expect from the

central limit theorem In probability theory, the central limit theorem (CLT) establishes that, in many situations, when independent random variables are summed up, their properly normalized sum tends toward a normal distribution even if the original variables themselv ...

References

{{Reflist Continuous distributions

See also

References