In probability theory and statistics, Student's distribution (or simply the distribution) is a continuous probability distribution that generalizes the standard normal distribution. Like the latter, it is symmetric around zero and bell-shaped.
However, has heavier tails, and the amount of probability mass in the tails is controlled by the parameter . For the Student's distribution becomes the standard Cauchy distribution, which has very "fat" tails; whereas for it becomes the standard normal distribution which has very "thin" tails.
The name "Student" is a pseudonym used by William Sealy Gosset in his scientific paper publications during his work at the Guinness Brewery in Dublin, Ireland.
The Student's distribution plays a role in a number of widely used statistical analyses, including Student's -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.
In the form of the location-scale distribution it generalizes the normal distribution and also arises in the Bayesian analysis of data from a normal family as a compound distribution when marginalizing over the variance parameter.
Definitions
Probability density function Student's distribution has the probability density function (PDF) given by
where is the number of degrees of freedom, and is the gamma function. This may also be written as
where is the beta function. In particular, for positive integer-valued degrees of freedom we have:
The probability density function is symmetric, and its overall shape 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 distribution approaches the normal distribution with mean 0 and variance 1. For this reason is also known as the normality parameter.
The following images show the density of the distribution for increasing values of The normal distribution is shown as a blue line for comparison. Note that the distribution (red line) becomes closer to the normal distribution as increases.
Cumulative distribution function The cumulative distribution function (CDF) can be written in terms of , the regularized incomplete beta function. For
where
Other values would be obtained by symmetry. An alternative formula, valid for is
where is a particular instance of the hypergeometric function.
For information on its inverse cumulative distribution function, see .
Special cases Certain values of give a simple form for Student's t-distribution. PDF CDF notes 1 See Cauchy distribution 2 3 4 5 See Normal distribution, Error function
Properties Moments For the raw moments of the distribution are
Moments of order or higher do not exist.
The term for even, may be simplified using the properties of the gamma function to
For a distribution with degrees of freedom, the expected value is if and its variance is if The skewness is 0 if and the excess kurtosis is if How the distribution arises (characterization)
As the distribution of a test statistic Student's t-distribution with degrees of freedom can be defined as the distribution of the random variable T with
where Z is a standard normal with expected value 0 and variance 1; V has a chi-squared distribution () with degrees of freedom; Z and V are independent;
A different distribution is defined as that of the random variable defined, for a given constant μ, by
This random variable has a noncentral t-distribution with noncentrality parameter μ. This distribution is important in studies of the power of Student's t-test.
Derivation Suppose X1, ..., Xn are independent realizations of the normally-distributed, random variable X, which has an expected value μ and variance σ2. Let
be the sample mean, and
be an unbiased estimate of the variance from the sample. It can be shown that the random variable
has a chi-squared distribution with 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 variance σ2/n. Moreover, it is possible to show that these two random variables (the normally distributed one Z and the chi-squared-distributed one V) are independent. Consequently the pivotal quantity
which differs from Z in that the exact standard deviation σ is replaced by the sample standard error s, 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 denominator, so it canceled. Gosset intuitively obtained the probability density function stated above, with equal to n − 1, and Fisher proved it in 1925.
The distribution of the test statistic T depends on , 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.
Sampling distribution of t-statistic The distribution arises as the sampling distribution of the statistic. Below the one-sample statistic is discussed, for the corresponding two-sample statistic see Student's t-test.
Unbiased variance estimate Let be independent and identically distributed samples from a normal distribution with mean and variance The sample mean and unbiased sample variance are given by:
The resulting (one sample) statistic is given by
and is distributed according to a Student's distribution with degrees of freedom.
Thus for inference purposes the statistic is a useful "pivotal quantity" in the case when the mean and variance are unknown population parameters, in the sense that the statistic has then a probability distribution that depends on neither nor
ML variance estimate Instead of the unbiased estimate we may also use the maximum likelihood estimate
yielding the statistic
This is distributed according to the location-scale distribution:
Compound distribution of normal with inverse gamma distribution The location-scale distribution results from compounding a Gaussian distribution (normal distribution) with mean and unknown variance, with an inverse gamma distribution placed over the variance with parameters and In other words, the random variable X is assumed to have a Gaussian distribution with an unknown variance distributed as inverse gamma, and then the variance is marginalized out (integrated out).
Equivalently, this distribution results from compounding a Gaussian distribution with a scaled-inverse-chi-squared distribution with parameters and The scaled-inverse-chi-squared distribution is exactly the same distribution as the inverse gamma distribution, but with a different parameterization, i.e.
The reason for the usefulness of this characterization is that in Bayesian statistics the inverse gamma distribution is the conjugate prior distribution of the variance of a Gaussian distribution. As a result, the location-scale distribution arises naturally in many Bayesian inference problems.
Maximum entropy distribution Student's distribution is the maximum entropy probability distribution for a random variate X having a certain value of . That is, for fixed >0, the Student's -distribution with degrees of freedom, , is the unique distribution that maximizes differential entropy subject to a fixed value of More precisely,
for some constant (the value of will depend on ), where denotes the set of all distributions on with finite differential entropy, and () is the differential entropy of . This follows immediately from the observation that the pdf can be written in exponential family form with as sufficient statistic.
Integral of Student's probability density function and -value The function is the integral of Student's probability density function, between and , for 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 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 tests. For the statistic , with degrees of freedom, is the probability that 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 It can be easily calculated from the cumulative distribution function of the distribution:
where is the regularized incomplete beta function.
For statistical hypothesis testing this function is used to construct the p-value.
Related distributions In general
The noncentral distribution generalizes the distribution to include a noncentrality parameter. Unlike the nonstandardized distributions, the noncentral distributions are not symmetric (the median is not the same as the mode). The discrete Student's distribution is defined by its probability mass function at r being proportional to: 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. One can generate Student samples by taking the ratio of variables from the normal distribution and the square-root of the . If we use instead of the normal distribution, e.g., the Irwin–Hall distribution, we obtain over-all a symmetric 4 parameter distribution, which includes the normal, the uniform, the triangular, the Student and the Cauchy distribution. This is also more flexible than some other symmetric generalizations of the normal distribution. distribution is an instance of ratio distributions. The square of a random variable distributed is distributed as Snedecor's F distribution .
Location-scale -distribution Location-scale transformation Student's distribution generalizes to the three parameter location-scale distribution by introducing a location parameter and a scale parameter With
and location-scale family transformation
we get
The resulting distribution is also called the non-standardized Student's distribution.
Density and first two moments The location-scale distribution has a density defined by:
Equivalently, the density can be written in terms of :
Other properties of this version of the distribution are:
Special cases If follows a location-scale distribution then for , is normally distributed with mean and variance The location-scale distribution with degree of freedom is equivalent to the Cauchy distribution The location-scale distribution with and reduces to the Student's distribution
Occurrence and applications In frequentist statistical inference Student's distribution arises in a variety of statistical estimation problems where the goal is to estimate an unknown parameter, such as a mean value, in a setting where the data are observed with additive errors. If (as in nearly all practical statistical work) the population standard deviation of these errors is unknown and has to be estimated from the data, the distribution is often used to account for the extra uncertainty that results from this estimation. In most such problems, if the standard deviation of the errors were known, a normal distribution would be used instead of the distribution.