Student's t-distribution

The Student 's t - distribution (including Student's t- distribution) is a probability distribution, which was developed in 1908 by William Sealy Gosset.

He had found that the standardized estimator of the sample mean value not normally distributed normally distributed data, but is t- distributed if the need for standardization of the mean variance of the trait is unknown and must be estimated using the sample variance. The t- distribution allows the calculation of the distribution of the difference from the sample mean to the true mean of the population. The t- values ​​depend on the significance level and n is the sample size and determine the confidence interval and thus the validity of the estimate of the mean. The t-distribution becomes narrower with increasing n and goes in for the normal distribution (see chart at right). Hypothesis testing, where the t distribution is used, is known as t-test.

The derivation was first published in 1908, while Gosset worked at a Guinness Brewery. Because his employer would not allow the publication, Gosset published it under the pseudonym Student. The t- factor and the associated theory were first occupied by the work of RA Fisher, who called the distribution of Student's distribution (Students distribution).

  • 5.1 Relationship to the Cauchy distribution
  • 5.2 Relationship to the distribution and the standard normal distribution
  • 5.3 approximated by the normal distribution
  • Table 9.1 some t- quantiles

Definition

A continuous random variable satisfies the Student's t-distribution with degrees of freedom when the probability density

For has. It is

The gamma function.

Alternatively, the t-distribution with n degrees of freedom defined as the distribution of the size

With a standard normal random variable, and is one of independent, χ ² - distributed random variable with degrees of freedom.

Distribution

The distribution function can be expressed as a closed

Or

With

Where the beta function is.

Calculates the probability that a random variable X distributed according to a value gets smaller or equal to t.

Properties

Let X be a t- distributed random variable with n degrees of freedom and density

Turning points

The density has turning points at

Median

The median is

Mode

The mode is given by

Expected value

For the expectation value is obtained for

The expected value for does not exist.

Variance

The variance is obtained for the

Skew

The skewness is

Curvatures

For the kurtosis excess camber and camber is obtained for

Moments

For the -th -th moments and the central moments applies:

Non- central t-distribution

If the counter of a t-distributed random variable normally distributed with an expected value, it is called a non-central t- distribution with non -centrality parameter. This distribution is used with t- distributed test statistic primarily for the determination of β - error in hypothesis testing.

Relationship to other distributions

Relationship with the Cauchy distribution

For and with the Cauchy distribution is obtained as a special case of the Student's t-distribution.

Relation to the distribution and the standard normal distribution

The t-distribution describes the distribution of an expression

With a standard normal and a χ ² - distributed random variable with degrees of freedom means. The counter variable must be independent of the denominator variable. The density of the t-distribution function is then symmetrical relative to their expected value 0 The values ​​of the distribution function are usually in the tabulated.

Approximated by the normal distribution

As the number of degrees of freedom one can approximate the distribution of values ​​of the t- distribution by the normal distribution. As a rule of thumb, that from 30 degrees of freedom, the t- distribution function can be approximated by the normal distribution.

Use in mathematical statistics

Various estimators are t- distributed.

When the independent random variables are normally distributed with the same mean and standard deviation, it can be proved that the sample mean

Because the random variable has a standard normal distribution, and a chi -square distribution with degrees of freedom follows, shows that the size

Is, by definition, t -distributed with n-1 degrees of freedom.

So the distance of the measured mean value is divided by the mean of the population as. Using it to calculate the 95 % confidence interval for the mean to

Where t is determined by. This interval is slightly larger than that which would have been obtained with known from the distribution function of the normal distribution with the same confidence level for.

Derivation of the density

The probability density of the t-distribution can be derived from the joint density of the two independent random variables Z and the standard normal, or chi-square distributed are.

The transformation

You get the joint density of and, where and.

The Jacobian of the transformation is:

The value is not important because it is multiplied by the calculation of the determinant to 0. The new density function is written like

Wanted is now the marginal distribution as an integral over the non- interest variable v:

Standard score Student's t-test Degrees of freedom (statistics) Moment (mathematics) Beta-binomial distribution Boltzmann distribution Uniform distribution (discrete) Geometric distribution Logarithmic distribution Zeta distribution Triangular distribution Beta prime distribution Bose–Einstein statistics Erlang distribution F-distribution Fermi–Dirac statistics Generalized extreme value distribution Hotelling's T-squared distribution Rayleigh distribution Weibull distribution Gumbel distribution Stable distribution Logistic distribution Multinomial distribution Dirichlet distribution Multivariate normal distribution Wishart distribution
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