Student distribution

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).

References

[1]	H. Cramér, "Mathematical methods of statistics" , Princeton Univ. Press (1946)
[2]	L.N. Bol'shev, N.V. Smirnov, "Tables of mathematical statistics" , Libr. math. tables , 46 , Nauka (1983) (In Russian) (Processed by L.S. Bark and E.S. Kedrova)
[3]	"Student" (W.S. Gosset), "The probable error of a mean" Biometrika , 6 (1908) pp. 1–25