Student's t-distribution From Wikipedia, the free encyclopedia Student's t In probability and statistics , Student's t- distribution (or simply the t-distribution ) is a Probability density function continuous probability distribution that arises when estimating the mean of a normally distributed population in situations where the sample size is small. It 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. The t-distribution is symmetric and bell -shaped, like the normal distribution, but has heavier tails, Cumulative distribution function 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 . Contents ■ 1 Introduction ■ 1.1 History and etymology parameters: ν > 0 ■ 1.2 Examples degrees of freedom (real) support: ■ 2 Characterization ■ 2.1 Probability density function pdf: ■ 2.1.1 Derivation ■ 2.2 Cumulative distribution function cdf: ■ 3 Properties ■ 3.1 Moments ■ 3.2 Related distributions ■ 3.3 Monte Carlo sampling where 2F1 is the hypergeometric ■ 3.4 Integral of Student's function probability density function and p- mean: 0 for ν > 1, otherwise undefined value ■ 4 Related distributions median: 0 ■ 4.1 Three-parameter version ■ 4.2 Discrete version mode: 0 ■ 5 Special cases variance: ■ 5.1 ν = 1 , for ■ 5.2 ν = 2 , otherwise undefined ■ 6 Uses skewness: 0 for ν > 3 ■ 6.1 In frequentist statistical inference ex.kurtosis: ■ 6.1.1 Hypothesis testing ■ 6.1.2 Confidence intervals entropy: ■ 6.1.3 Prediction intervals ■ 6.2 Robust parametric modeling ■ 7 Table of selected values ■ 8 See also ■ ψ: digamma function, ■ 9 Notes ■ B: beta function ■ 10 References ■ 11 External links mgf: (Not defined) cf: Introduction [1] History and etymology ■ Kν(x): Bessel function In statistics, the t-distribution was first derived as a posterior distribution by Helmert and Lüroth.[2][3][4] In the English literature, a derivation of the t-distribution was published in 1908 by William Sealy Gosset [5] while he worked at the Guinness Brewery in Dublin. Since Gosset's employer forbade members of its staff from publishing scientific papers, his work was published under the pseudonym Student . The t-test and the associated theory became well-known through the work of R.A. Fisher, who called the distribution "Student's distribution". [6][7] Examples For examples of the use of this distribution, see Student's t test . Characterization Student's t-distribution is the probability distribution of the ratio [8] where ■ Z is normally distributed with expected value 0 and variance 1; ■ V has a chi-square distribution with ν ("nu") degrees of freedom; ■ Z and V are independent. While, for any given constant µ, is a random variable of noncentral t-distribution with noncentrality parameter µ. Probability density function Student's t-distribution has the probability density function where ν is the number of degrees of freedom and Γ is the Gamma function. For ν even, For ν odd, The overall shape of the probability density function of the t-distribution resembles the bell shape of a normally distributed variable with mean 0 and variance 1, except that it is a bit lower and wider. As the number of degrees of freedom grows, the t-distribution approaches the normal distribution with mean 0 and variance 1. The following images show the density of the t-distribution for increasing values of ν. The normal distribution is shown as a blue line for comparison. Note that the t-distribution (red line) becomes closer to the normal distribution as ν increases. Density of the t-distribution (red) for 1, 2, 3, 5, 10, and 30 df compared to normal distribution (blue). Previous plots shown in green. 1 degree of freedom 2 degrees of freedom 3 degrees of freedom 5 degrees of freedom 10 degrees of freedom 30 degrees of freedom Derivation Suppose X1, ..., Xn are independent values that are normally distributed with expected value µ and variance σ2. Let be the sample mean, and be the sample variance. It can be shown that the random variable has a chi -square distribution with n − 1 degrees of freedom (by Cochran's theorem ). It is readily shown that the quantity is normally distributed with mean 0 and variance 1, since the sample mean is normally distributed with mean μ and standard error . Moreover, it is possible to show that these two random variables—the normally distributed one and the chi-square-distributed one—are independent. Consequently the pivotal quantity , which differs from Z in that the exact standard deviation σ is replaced by the random variable Sn, has a Student's t-distribution as defined above. Notice that the unknown population variance σ2 does not appear in T, since it was in both the numerator and the denominators, so it canceled. Gosset's work showed that T has the probability density function with ν equal to n − 1. This may also be written as where B is the Beta function . The distribution of T is now called the t-distribution . The parameter ν is called the number of degrees of freedom . The distribution depends on ν, but not µ or σ; the lack of dependence on µ and σ is what makes the t-distribution important in both theory and practice. Gosset's result can be stated more generally. (See, for example, Hogg and Craig, Sections 4.4 and 4.8.) Let Z have a normal distribution with mean 0 and variance 1. Let V have a chi-square distribution with ν degrees of freedom. Further suppose that Z and V are independent (see Cochran's theorem). Then the ratio has a t-distribution with ν degrees of freedom. Cumulative distribution function The cumulative distribution function is given by the regularized incomplete beta function , with Properties Moments The moments of the t -distribution are It should be noted that the term for 0 < k < ν, k even, may be simplified using the properties of the Gamma function to For a t-distribution with ν degrees of freedom, the expected value is 0, and its variance is ν/( ν − 2) if ν > 2. The skewness is 0 if ν > 3 and the excess kurtosis is 6/( ν − 4) if ν > 4. Related distributions ■ has a t-distribution if has a scaled inverse-χ2 distribution and has a normal distribution. ■ has an F-distribution if and has a Student's t- distribution. Monte Carlo sampling There are various approaches to constructing random samples from the Student distribution. The matter depends on whether the samples are required on a stand-alone basis, or are to be constructed by application of a quantile function to uniform samples, e.g. in multi-dimensional applications basis on copula-dependency. In the case of stand-alone sampling, Bailey's 1994 extension of the Box-Muller method and its polar variation are easily deployed. It has the merit that it applies equally well to all real positive and negative degrees of freedom. Integral of Student's probability density function and p-value The function is the integral of Student's probability density function, ƒ(t) between −t and t. It thus gives the probability that a value of t less than that calculated from observed data would occur by chance. Therefore, the function can be used when testing whether the difference between the means of two sets of data is statistically significant, by calculating the corresponding value of t and the probability of its occurrence if the two sets of data were drawn from the same population. This is used in a variety of situations, particularly in t-tests. For the statistic t, with degrees of freedom, is the probability that t would be less than the observed value if the two means were the same (provided that the smaller mean is subtracted from the larger, so that t > 0). It is defined for real t by the following formula: where B is the Beta function . For t > 0, there is a relation to the regularized incomplete beta function Ix (a, b) as follows: For statistical hypothesis testing this function is used to construct the p-value . Related distributions Three -parameter version Student's t distribution can be generalized to a three parameter location/scale family [9] that introduces a location parameter μ and an inverse scale parameter (i.e. precision) λ, and has a density defined by Other properties of this version of the distribution are [9]: This distribution results from compounding a Gaussian distribution with mean μ and unknown precision (the reciprocal of the variance), with a gamma distribution with parameters a = ν / 2 and b = ν / 2λ. In other words, the random variable X is assumed to have a normal distribution with an unknown precision distributed as gamma, and then this is marginalized over the gamma distribution. (The reason for the usefulness of this characterization is that the gamma distribution is the conjugate prior distribution of the precision of a Gaussian distribution. As a result, the three-parameter Student's t distribution arises naturally in many Bayesian inference problems.) The noncentral t -distribution is a different way of generalizing the t-distribution to include a location parameter.