Task: Question Answering
Input (English): Context: 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 : by supplying a valid probability mass function or probability density function by supplying a valid cumulative distribution function or survival function by supplying a valid hazard function by supplying a valid characteristic function by supplying a rule for constructing a new random variable from other random variables whose joint probability distribution is known . A probability distribution can either be univariate or multivariate . A univariate distribution gives the probabilities of a single random variable taking on various alternative values ; a multivariate distribution ( a joint probability distribution ) gives the probabilities of a random vector -- a set of two or more random variables -- taking on various combinations of values . Important and commonly encountered univariate probability distributions include the binomial distribution , the hypergeometric distribution , and the normal distribution . The multivariate normal distribution is a commonly encountered multivariate distribution ., Analysis is the process of breaking a complex topic or substance into smaller parts in order to gain a better understanding of it. The technique has been applied in the study of mathematics and logic since before Aristotle (384322 B.C.), though "analysis" as a formal concept is a relatively recent development., An experiment is a procedure carried out to support, refute, or validate a hypothesis. Experiments provide insight into cause-and-effect by demonstrating what outcome occurs when a particular factor is manipulated. Experiments vary greatly in goal and scale, but always rely on repeatable procedure and logical analysis of the results. There also exists natural experimental studies., Probability is the measure of the likelihood that an event will occur. Probability is quantified as a number between 0 and 1 (where 0 indicates impossibility and 1 indicates certainty). The higher the probability of an event, the more certain that the event will occur. A simple example is the tossing of a fair (unbiased) coin. Since the coin is unbiased, the two outcomes ("head" and "tail") are both equally probable; the probability of "head" equals the probability of "tail". Since no other outcomes are possible, the probability is 1/2 (or 50%), of either "head" or "tail". In other words, the probability of "head" is 1 out of 2 outcomes and the probability of "tail" is also 1 out of 2 outcomes, expressed as 0.5 when converted to decimal, with the above-mentioned quantification system. This type of probability is also called a priori probability., In statistics, a population is a set of similar items or events which is of interest for some question or experiment. A statistical population can be a group of actually existing objects (e.g. the set of all stars within the Milky Way galaxy) or a hypothetical and potentially infinite group of objects conceived as a generalization from experience (e.g. the set of all possible hands in a game of poker). A common aim of statistical analysis is to produce information about some chosen population., Debt, AIDS, Trade, Africa (DATA) was a multinational non-government organization founded in January 2002 in London by U2's Bono along with Bobby Shriver and activists from the Jubilee 2000 Drop the Debt campaign., Probability and Statistics or also called Statistics and Probability are two related but separate academic disciplines. Statistical analysis often uses probability distributions, and the two topics are often studied together. However, probability theory contains much that is mostly of mathematical interest and not directly relevant to statistics. Moreover, many topics in statistics are independent of probability theory., A statistic (singular) or sample statistic is a single measure of some attribute of a sample (e.g., its arithmetic mean value). It is calculated by applying a function (statistical algorithm) to the values of the items of the sample, which are known together as a set of data., A statistical model is a class of mathematical model, which embodies a set of assumptions concerning the generation of some sample data, and similar data from a larger population. A statistical model represents, often in considerably idealized form, the data-generating process., In probability and statistics, a random variable, random quantity, aleatory variable, or stochastic variable is a variable quantity whose possible values depend, in random manner, on a set of random outcomes events. It is common that the outcome depends on some physical variables that are not well understood. For example,
when you toss a coin (or a die), it falls on its edge, and the final outcome formula_1 or formula_2
depends on the uncertain physics. What is not uncertain are the possible outcomes. Of course
the coin could get caught in a crack in the floor, but that is not a consideration. Essential to the definition of a random variable is the outcome space typically defined in terms of set theory or a set. In the case of the coin, the set is binary.
The member of the set are each labelled by a probability formula_3 where formula_4 labels the possible outcome values. Thus for the coin example,
there are formula_5 such that formula_6. The only outcomes allowed are those from the set, thus one
of these outcomes must have some measurable (non-zero) probability., Statistics is the study of the collection, analysis, interpretation, presentation, and organization of data. In applying statistics to, e.g., a scientific, industrial, or social problem, it is conventional to begin with a statistical population or a statistical model process to be studied. Populations can be diverse topics such as "all people living in a country" or "every atom composing a crystal". Statistics deals with all aspects of data including the planning of data collection in terms of the design of surveys and experiments., An "a priori" probability is a probability that is derived purely by deductive reasoning. One way of deriving "a priori" probabilities is the principle of indifference, which has the character of saying that, if there are "N" mutually exclusive and exhaustive events and if they are equally likely, then the probability of a given event occurring is 1/"N". Similarly the probability of one of a given collection of "K" events is "K"/"N"., Subject: probability distribution, Relation: subclass_of, Options: (A) analysis (B) attribute (C) c (D) campaign (E) character (F) class (G) coin (H) concept (I) consideration (J) country (K) course (L) data (M) data collection (N) design (O) die (P) edge (Q) event (R) experiment (S) fair (T) final (U) galaxy (V) game (W) government (X) group (Y) information (Z) mathematical model ([) measure (\) model (]) natural (^) number (_) order (`) organization (a) out (b) physics (c) principle (d) quantity (e) reasoning (f) single (g) statistic (h) statistics (i) study (j) theory (k) trade (l) value (m) variable
Output (English): mathematical model