Student's t-distribution - York University

[Pages:15]Student's t-distribution

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In probability and statistics, Student's tdistribution (or simply the t-distribution) is a 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 tdistribution 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, 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 tdistribution is a special case of the generalised hyperbolic distribution.

Student's t Probability density function

Cumulative distribution function

Contents

1 Introduction 1.1 History and etymology 1.2 Examples

2 Characterization 2.1 Probability density function 2.1.1 Derivation

2.2 Cumulative distribution function

3 Properties 3.1 Moments 3.2 Related distributions 3.3 Monte Carlo sampling 3.4 Integral of Student's probability density function and pvalue

parameters: > 0 degrees of freedom (real)

support: pdf:

cdf:

mean:

where 2F1 is the hypergeometric function

0 for > 1, otherwise undefined

 4 Related distributions 4.1 Three-parameter version 4.2 Discrete version

5 Special cases 5.1 = 1 5.2 = 2

6 Uses 6.1 In frequentist statistical inference 6.1.1 Hypothesis testing 6.1.2 Confidence intervals 6.1.3 Prediction intervals

6.2 Robust parametric modeling

7 Table of selected values 8 See also 9 Notes 10 References 11 External links

Introduction

History and etymology

median: 0

mode: 0

variance:

, for

, otherwise undefined

skewness: 0 for > 3

ex.kurtosis:

entropy:

mgf: cf:

: digamma function, B: beta function

(Not defined)

K(x): Bessel function[1]

In statistics, the t-distribution was first derived as a posterior distribution by Helmert and L?roth.[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 ("nu") degrees of freedom;

Z and V are independent.

While, for any given constant ?,

is a random variable of noncentral t-distribution with

noncentrality parameter ?.

Probability density function

Student's t-distribution has the probability density function

where is the number of degrees of freedom and is the Gamma function. For even,

For 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 . 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 increases.

Density of the t-distribution (red) for 1, 2, 3, 5, 10, and 30 df compared to normal distribution (blue). Previous plot 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 ? and variance 2. 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 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 is replaced by the random variable Sn, has a Student's t-distribution as defined above. Notice that the unknown population variance 2 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 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 is called the number of degrees of freedom. The distribution depends on , but not ? or ; the lack of dependence on ? and 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

degrees of freedom. Further suppose that Z and V are independent (see Cochran's theorem). Then the ratio

has a t-distribution with 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 < , k even, may be simplified using the properties of the

Gamma function to

For a t-distribution with degrees of freedom, the expected value is 0, and its variance is /( - 2) if > 2. The skewness is 0 if > 3 and the excess kurtosis is 6/( - 4) if > 4.

Related distributions

has a t-distribution if

has a normal distribution.

has an F-distribution if

distribution.

has a scaled inverse-2 distribution and

and

has a Student's t-

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, (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 and an inverse scale parameter (i.e. precision) , 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 and unknown precision (the reciprocal of the variance), with a gamma distribution with parameters a = / 2 and b = / 2. 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.

Discrete version

The "discrete Student's t distribution" is defined by its probability mass function at r being proportional to[10]

Here 'a', b, and k are parameters. This distribution arises from the construction of a system of discrete distributions similar to that of the Pearson distributions for continuous distributions.[11]

Special cases

Certain values of give an especially simple form. = 1

Distribution function:

Density function:

See Cauchy distribution

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