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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Student's t-distribution ： ウィキペディア英語版 Student's t-distribution \!| cdf =$\backslash begin$ \frac + x \Gamma \left( \frac \right) \times\\() \frac,\frac;\frac; -\frac \right)} \right)} \end >where 2''F''1 is the hypergeometric function| mean =0 for $\backslash nu\; >\; 1$, otherwise undefined| median =0| mode =0| variance =$\backslash textstyle\backslash frac$ for $\backslash nu\; >\; 2$, ∞ for , otherwise undefined| skewness =0 for $\backslash nu\; >\; 3$, otherwise undefined| kurtosis =$\backslash textstyle\backslash frac$ for $\backslash nu\; >\; 4$, ∞ for , otherwise undefined| entropy =$\backslash begin$ \frac\left( \psi \left(\frac \right) - \psi \left(\frac \right) \right ) \\() + \logB">\left(\frac,\frac \right)\right )} \end * ψ: digamma function, * ''B'': beta function| | mgf = undefined| char =$\backslash textstyle\backslash frac|t|\backslash right)$ \cdot \left(\sqrt|t| \right)^} } In probability and statistics, Student's ''t''-distribution (or simply the ''t''-distribution) is any member of a family of continuous probability distributions that arises when estimating the mean of a normally distributed population in situations where the sample size is small and population standard deviation is unknown developed by William Sealy Gosset under the pseudonym ''Student''. Whereas a normal distribution describes a full population, ''t''-distributions describe samples drawn from a full population; accordingly, the ''t''-distribution for each sample size is different, and the larger the sample, the more the distribution resembles a normal distribution. The ''t''-distribution plays a role in a number of widely used statistical analyses, including the Student's ''t''-test for assessing the statistical significance of the difference between two sample means, the construction of confidence intervals for the difference between two population means, and in linear regression analysis. The Student's ''t''-distribution also arises in the Bayesian analysis of data from a normal family. If we take a sample of ''n'' observations from a normal distribution, then the ''t''-distribution with $\backslash nu=n-1$ degrees of freedom can be defined as the distribution of the location of the true mean, relative to the sample mean and divided by the sample standard deviation, after multiplying by the normalizing term $\backslash sqrt$. In this way, the ''t''-distribution can be used to estimate how likely it is that the true mean lies in any given range. The ''t''-distribution is symmetric and bell-shaped, like the normal distribution, but has heavier tails, meaning that it is more prone to producing values that fall far from its mean. This makes it useful for understanding the statistical behavior of certain types of ratios of random quantities, in which variation in the denominator is amplified and may produce outlying values when the denominator of the ratio falls close to zero. The Student's ''t''-distribution is a special case of the generalised hyperbolic distribution. ==History and etymology== In statistics, the ''t''-distribution was first derived as a posterior distribution in 1876 by Helmert〔〔〔 and Lüroth.〔 In the English-language literature it takes its name from William Sealy Gosset's 1908 paper in ''Biometrika'' under the pseudonym "''Student''".〔"Student" (William Sealy Gosset), original Biometrika paper as a (scan )〕 Gosset worked at the Guinness Brewery in Dublin, Ireland, and was interested in the problems of small samples, for example the chemical properties of barley where sample sizes might be as low as 3. One version of the origin of the pseudonym is that Gosset's employer preferred staff to use pen names when publishing scientific papers instead of their real name, so he used the name "''Student''" to hide his identity. Another version is that Guinness did not want their competitors to know that they were using the ''t''-test to test the quality of raw material.〔Mortimer, Robert G. (2005) ''Mathematics for Physical Chemistry'', Academic Press. 3 edition. ISBN 0-12-508347-5 (page 326)〕 Gosset's paper refers to the distribution as the "frequency distribution of standard deviations of samples drawn from a normal population". It became well-known through the work of Ronald Fisher, who called the distribution "''Students distribution" and referred to the value as ''t''.〔Walpole, Ronald; Myers, Raymond; Myers, Sharon; Ye, Keying. (2002) ''Probability and Statistics for Engineers and Scientists''. Pearson Education, 7th edition, pg. 237 ISBN 81-7758-404-9〕 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Student's t-distribution」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.