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In probability theory and statistics, Student's t distribution (or simply the t distribution) is a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped.
Most frequently, t statistics are used in Student's t-tests, a form of statistical hypothesis testing, and in the computation of certain confidence intervals. The key property of the t statistic is that it is a pivotal quantity – while defined in terms of the sample mean, its sampling distribution does not depend on the population parameters, and thus it can be used regardless of what these ...
The term "t-statistic" is abbreviated from "hypothesis test statistic". [1] In statistics, the t-distribution was first derived as a posterior distribution in 1876 by Helmert [2] [3] [4] and Lüroth. [5] [6] [7] The t-distribution also appeared in a more general form as Pearson type IV distribution in Karl Pearson's 1895 paper. [8]
Probability 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 .
Probability is the branch of mathematics and statistics concerning events and numerical descriptions of how likely they are to occur. The probability of an event is a number between 0 and 1; the larger the probability, the more likely an event is to occur. [note 1] [1] [2] This number is often expressed as a percentage (%), ranging from 0% to ...
The noncentral t-distribution generalizes Student's t-distribution using a noncentrality parameter.Whereas the central probability distribution describes how a test statistic t is distributed when the difference tested is null, the noncentral distribution describes how t is distributed when the null is false.
The probability is sometimes written to distinguish it from other functions and measure P to avoid having to define "P is a probability" and () is short for ({: ()}), where is the event space, is a random variable that is a function of (i.e., it depends upon ), and is some outcome of interest within the domain specified by (say, a particular ...
A discrete probability distribution is applicable to the scenarios where the set of possible outcomes is discrete (e.g. a coin toss, a roll of a die) and the probabilities are encoded by a discrete list of the probabilities of the outcomes; in this case the discrete probability distribution is known as probability mass function.