Probability distribution

In probability and statistics, a probability distribution assigns a probability to each measurable subset of the possible outcomes of a random experiment, survey, or procedure of statistical inference. Examples are found in experiments whose sample space is non-numerical, where the distribution would be a categorical distribution; experiments whose sample space is encoded by discrete random variables, where the distribution can be specified by a probability mass function; and experiments with sample spaces encoded by continuous random variables, where the distribution can be specified by a probability density function. More complex experiments, such as those involving stochastic processes defined in continuous time, may demand the use of more general probability measures.

In applied probability, a probability distribution can be specified in a number of different ways, often chosen for mathematical convenience:

History monoid

In mathematics and computer science, a history monoid is a way of representing the histories of concurrently running computer processes as a collection of strings, each string representing the individual history of a process. The history monoid provides a set of synchronization primitives (such as locks, mutexes or thread joins) for providing rendezvous points between a set of independently executing processes or threads.

History monoids occur in the theory of concurrent computation, and provide a low-level mathematical foundation for process calculi, such as CSP the language of communicating sequential processes, or CCS, the calculus of communicating systems. History monoids were first presented by M.W. Shields.

History monoids are isomorphic to trace monoids (free partially commutative monoids) and to the monoid of dependency graphs. As such, they are free objects and are universal. The history monoid is a type of semi-abelian categorical product in the category of monoids.

Distribution (number theory)

In algebra and number theory, a distribution is a function on a system of finite sets into an abelian group which is analogous to an integral: it is thus the algebraic analogue of a distribution in the sense of generalised function.

The original examples of distributions occur, unnamed, as functions φ on Q/Z satisfying

We shall call these ordinary distributions. They also occur in p-adic integration theory in Iwasawa theory.

Let ... → X_n+1 → X_n → ... be a projective system of finite sets with surjections, indexed by the natural numbers, and let X be their projective limit. We give each X_n the discrete topology, so that X is compact. Let φ = (φ_n) be a family of functions on X_n taking values in an abelian group V and compatible with the projective system:

for some weight function w. The family φ is then a distribution on the projective system X.

A function f on X is "locally constant", or a "step function" if it factors through some X_n. We can define an integral of a step function against φ as

Star (heraldry)

In heraldry, the term star may refer to any star-shaped charge with any number of rays, which may appear straight or wavy, and may or may not be pierced. While there has been much confusion between the two due to their similar shape, a star with straight-sided rays is usually called a mullet while one with wavy rays is usually called an estoile.

While a mullet may have any number of points, it is presumed to have five unless otherwise specified in the blazon, and pierced mullets are common; estoiles, however, are presumed to have six rays and (as of 1909) had not been found pierced. In Scottish heraldry, an estoile is the same as in English heraldry, but it has been said that mullet refers only to a mullet pierced (also called a spur revel), while one that is not pierced is called a star.

Terminology

The use of the word star in blazons, and how that charge appears in coat armory, varies from one jurisdiction to another. In Scots heraldry, both star and mullet interchangeably mean a star with five straight rays; the official record from 1673 gives Murray of Ochtertyre azur three Starrs argent ... (Public Register, vol 1 p 188), while the Ordinary of Arms produced by a late 19th century Lyon King of Arms 'modernizes' the original as Az. three mullets arg. .... In Canadian heraldry the usual term is mullet, but there is also the occasional six-pointed star (e.g. in Vol. IV, at p. 274 and in online version of the Canadian Public Register), which is what others would blazon as a six-pointed mullet. The United States Army Institute of Heraldry, the official heraldic authority in the United States, uses the term mullet in its blazons, but elsewhere, as in US government documents describing the flag of the United States and the Great Seal of the United States, the term star is constantly used, and these nearly always appear with five straight-sided points.

Celebrity

Celebrity is fame and public attention in the media, usually applied to a person, or group of people (celebrity couple, family etc.), or occasionally, to animals or fictional entities. Celebrity status is often associated with wealth (commonly referred to as fame and fortune) and fame can often provide opportunities to make money.

Successful careers in sports and entertainment are commonly associated with celebrity status; political leaders often become celebrities. People may also become celebrities due to media attention for their lifestyle, wealth, or controversial actions, or for their connection to a famous person.

History

Throughout recorded history there are accounts of people who attracted the trappings of celebrity which would be recognized today.

Athletes in Ancient Greece were welcomed home as heroes, had songs and poems written in their honour and received free food and gifts from those seeking celebrity endorsement.Ancient Rome similarly lauded actors and notorious gladiators and Julius Caesar appeared on a coin in his own lifetime (a departure from the usual depiction of battles and divine lineage).

Star (702 album)

Star is the third and final studio album from American R&B group 702, released March 25, 2003 by Motown.

The album peaked at number forty-five on the Billboard 200 chart. and is mostly remembered for its cult classic single "I Still Love You".

Release and reception

The album peaked at forty-five on the U.S. Billboard 200 and reached the twenty-second spot on the R&B Albums chart.

Andy Kellman of Allmusic gave the work a rather dismissive review, stating that "it continues in the group's tradition of being able to deliver a couple of solid singles surrounded by middling to fair album tracks."

Track listing

Chart history

Album

Singles

Personnel

Information taken from Allmusic.