Difference between revisions of "Student distribution"
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907105.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907105.png" /></td> </tr></table> | ||
− | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907106.png" /> is a random variable subject to the standard normal law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907108.png" /> is a random variable not depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907109.png" /> and subject to the [[ | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907106.png" /> is a random variable subject to the standard normal law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907107.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907108.png" /> is a random variable not depending on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907109.png" /> and subject to the [[Chi-squared distribution| "chi-squared" distribution]] with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071010.png" /> degrees of freedom. The distribution function of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071011.png" /> is expressed by the formula |
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071012.png" /></td> </tr></table> | <table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071012.png" /></td> </tr></table> |
Revision as of 11:51, 20 October 2012
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 |
Student distribution. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Student_distribution&oldid=15207