# Student distribution

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with degrees of freedom, -distribution

The probability distribution of the random variable where is a random variable subject to the standard normal law and is a random variable not depending on and subject to the "chi-squared" distribution with degrees of freedom. The distribution function of the random variable is expressed by the formula  In particular, if , then is the distribution function of the Cauchy distribution. The probability density of the Student distribution is symmetric about 0, therefore The moments of a Student distribution exist only for , the odd moments are equal to 0, and, in particular . The even moments of a Student distribution are expressed by the formula in particular, . The distribution function of the random variable is expressed in terms of the beta-distribution function in the following way: where is the incomplete beta-function, . If , then the Student distribution converges to the standard normal law, i.e. Example. Let be independent, identically, normally -distributed random variables, where the parameters and are unknown. Then the statistics are the best unbiased estimators of and ; here and are stochastically independent. Since the random variable is subject to the standard normal law, while is distributed according to the "chi-squared" law with degrees of freedom, then by virtue of their independence, the fraction is subject to the Student distribution with degrees of freedom. Let and be the solutions of the equations Then the statistics and are the lower and upper bounds of the confidence set for the unknown mathematical expectation of the normal law , and the confidence coefficient of this confidence set is equal to , i.e. The Student distribution was first used by W.S. Gosset (pseudonym Student).

How to Cite This Entry:
Student distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Student_distribution&oldid=15207
This article was adapted from an original article by M.S. Nikulin (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article