Student distribution
with degrees of freedom,
-distribution
The probability distribution of the random variable
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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
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In particular, if , then
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is the distribution function of the Cauchy distribution. The probability density of the Student distribution is symmetric about 0, therefore
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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
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in particular, . The distribution function
of the random variable
is expressed in terms of the beta-distribution function in the following way:
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where is the incomplete beta-function,
. If
, then the Student distribution converges to the standard normal law, i.e.
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Example. Let be independent, identically, normally
-distributed random variables, where the parameters
and
are unknown. Then the statistics
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are the best unbiased estimators of and
; here
and
are stochastically independent. Since the random variable
is subject to the standard normal law, while
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is distributed according to the "chi-squared" law with degrees of freedom, then by virtue of their independence, the fraction
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is subject to the Student distribution with degrees of freedom. Let
and
be the solutions of the equations
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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.
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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 |
Student distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Student_distribution&oldid=15207