MarathonThe marathon is a long-distance foot race with a distance of , usually run as a road race, but the distance can be covered on trail routes. The marathon can be completed by running or with a run/walk strategy. There are also wheelchair divisions. More than 800 marathons are held throughout the world each year, with the vast majority of competitors being recreational athletes, as larger marathons can have tens of thousands of participants. The marathon was one of the original modern Olympic events in 1896.
ProbabilityProbability is the branch of mathematics concerning numerical descriptions of how likely an 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 1, where, roughly speaking, 0 indicates impossibility of the event and 1 indicates certainty. The higher the probability of an event, the more likely it is that the event will occur. A simple example is the tossing of a fair (unbiased) coin.
Probability theoryProbability theory or probability calculus 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 of axioms. Typically these axioms formalise probability in terms of a probability space, which assigns a measure taking values between 0 and 1, termed the probability measure, to a set of outcomes called the sample space.
Conditional probabilityIn probability theory, conditional probability is a measure of the probability of an event occurring, given that another event (by assumption, presumption, assertion or evidence) has already occurred. This particular method relies on event B occurring with some sort of relationship with another event A. In this event, the event B can be analyzed by a conditional probability with respect to A. If the event of interest is A and the event B is known or assumed to have occurred, "the conditional probability of A given B", or "the probability of A under the condition B", is usually written as P(AB) or occasionally P_B(A).
Probability distributionIn 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 in terms of its sample space and the probabilities of events (subsets of the sample space). For instance, if X is used to denote the outcome of a coin toss ("the experiment"), then the probability distribution of X would take the value 0.5 (1 in 2 or 1/2) for X = heads, and 0.
Probability axiomsThe Kolmogorov axioms are the foundations of probability theory introduced by Russian mathematician Andrey Kolmogorov in 1933. These axioms remain central and have direct contributions to mathematics, the physical sciences, and real-world probability cases. An alternative approach to formalising probability, favoured by some Bayesians, is given by Cox's theorem. The assumptions as to setting up the axioms can be summarised as follows: Let be a measure space with being the probability of some event , and .
Probability spaceIn probability theory, a probability space or a probability triple is a mathematical construct that provides a formal model of a random process or "experiment". For example, one can define a probability space which models the throwing of a die. A probability space consists of three elements: A sample space, , which is the set of all possible outcomes. An event space, which is a set of events, , an event being a set of outcomes in the sample space. A probability function, , which assigns each event in the event space a probability, which is a number between 0 and 1.
Bayesian probabilityBayesian probability (ˈbeɪziən or ˈbeɪʒən ) is an interpretation of the concept of probability, in which, instead of frequency or propensity of some phenomenon, probability is interpreted as reasonable expectation representing a state of knowledge or as quantification of a personal belief. The Bayesian interpretation of probability can be seen as an extension of propositional logic that enables reasoning with hypotheses; that is, with propositions whose truth or falsity is unknown.
Frequentist probabilityFrequentist probability or frequentism is an interpretation of probability; it defines an event's probability as the limit of its relative frequency in many trials (the long-run probability). Probabilities can be found (in principle) by a repeatable objective process (and are thus ideally devoid of opinion). The continued use of frequentist methods in scientific inference, however, has been called into question. The development of the frequentist account was motivated by the problems and paradoxes of the previously dominant viewpoint, the classical interpretation.
Probability interpretationsThe word probability has been used in a variety of ways since it was first applied to the mathematical study of games of chance. Does probability measure the real, physical, tendency of something to occur, or is it a measure of how strongly one believes it will occur, or does it draw on both these elements? In answering such questions, mathematicians interpret the probability values of probability theory. There are two broad categories of probability interpretations which can be called "physical" and "evidential" probabilities.
Long-distance runningLong-distance running, or endurance running, is a form of continuous running over distances of at least . Physiologically, it is largely aerobic in nature and requires stamina as well as mental strength. Within endurance running comes two different types of respiration. The more prominent side that runners experience more frequently is aerobic respiration. This occurs when oxygen is present, and the body can utilize oxygen to help generate energy and muscle activity.
Race and healthRace and health refers to how being identified with a specific race influences health. Race is a complex concept that has changed across chronological eras and depends on both self-identification and social recognition. In the study of race and health, scientists organize people in racial categories depending on different factors such as: phenotype, ancestry, social identity, genetic makeup and lived experience. "Race" and ethnicity often remain undifferentiated in health research.
Exponential distributionIn probability theory and statistics, the exponential distribution or negative exponential distribution is the probability distribution of the time between events in a Poisson point process, i.e., a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson point processes it is found in various other contexts.
Race and geneticsResearchers have investigated the relationship between race and genetics as part of efforts to understand how biology may or may not contribute to human racial categorization. Many constructions of race are associated with phenotypical traits and geographic ancestry, and scholars like Carl Linnaeus have proposed scientific models for the organization of race since at least the 18th century. Following the discovery of Mendelian genetics and the mapping of the human genome, questions about the biology of race have often been framed in terms of genetics.
Poisson distributionIn probability theory and statistics, the Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space if these events occur with a known constant mean rate and independently of the time since the last event. It is named after French mathematician Siméon Denis Poisson ('pwɑːsɒn; pwasɔ̃). The Poisson distribution can also be used for the number of events in other specified interval types such as distance, area, or volume.
Race (human categorization)Race is a categorization of humans based on shared physical or social qualities into groups generally viewed as distinct within a given society. The term came into common usage during the 16th century, when it was used to refer to groups of various kinds, including those characterized by close kinship relations. By the 17th century, the term began to refer to physical (phenotypical) traits, and then later to national affiliations. Modern science regards race as a social construct, an identity which is assigned based on rules made by society.
Critical race theoryCritical race theory (CRT) is an interdisciplinary academic field devoted to analysing how laws, social and political movements, and media shape, and are shaped by, social conceptions of race and ethnicity. CRT also considers racism to be systemic in various laws and rules, and not only based on individuals' prejudices. The word critical in the name is an academic reference to critical thinking, critical theory, and scholarly criticism, rather than criticizing or blaming individuals.
Mode (statistics)The mode is the value that appears most often in a set of data values. If X is a discrete random variable, the mode is the value x at which the probability mass function takes its maximum value (i.e, x=argmaxxi P(X = xi)). In other words, it is the value that is most likely to be sampled. Like the statistical mean and median, the mode is a way of expressing, in a (usually) single number, important information about a random variable or a population.
