Difference between revisions of "Student distribution"
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− | < | + | ''with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907102.png" /> degrees of freedom, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s0907104.png" />-distribution'' |
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− | degrees of freedom, | ||
− | distribution'' | ||
The probability distribution of the random variable | The probability distribution of the random variable | ||
− | + | <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> | |
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− | + | 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 | |
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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/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/s09071013.png" /></td> </tr></table> | |
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− | In particular, if | + | In particular, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071014.png" />, then |
− | then | ||
− | + | <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/s09071015.png" /></td> </tr></table> | |
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is the distribution function of the [[Cauchy distribution|Cauchy distribution]]. The probability density of the Student distribution is symmetric about 0, therefore | is the distribution function of the [[Cauchy distribution|Cauchy distribution]]. The probability density of the Student distribution is symmetric about 0, therefore | ||
− | + | <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/s09071016.png" /></td> </tr></table> | |
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− | + | The moments <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071017.png" /> of a Student distribution exist only for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071018.png" />, the odd moments are equal to 0, and, in particular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071019.png" />. The even moments of a Student distribution are expressed by the formula | |
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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/s09071020.png" /></td> </tr></table> | |
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− | + | in particular, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071021.png" />. The distribution function <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071022.png" /> of the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071023.png" /> is expressed in terms of the [[Beta-distribution|beta-distribution]] function in the following way: | |
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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/s09071024.png" /></td> </tr></table> | |
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− | + | where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071025.png" /> is the incomplete beta-function, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071027.png" />, then the Student distribution converges to the standard normal law, i.e. | |
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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/s09071028.png" /></td> </tr></table> | |
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− | + | Example. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071029.png" /> be independent, identically, normally <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071030.png" />-distributed random variables, where the parameters <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071031.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071032.png" /> are unknown. Then the statistics | |
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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/s09071033.png" /></td> </tr></table> | |
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− | + | are the best unbiased estimators of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071034.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071035.png" />; here <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071036.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071037.png" /> are stochastically independent. Since the random variable <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071038.png" /> is subject to the standard normal law, while | |
− | + | <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/s09071039.png" /></td> </tr></table> | |
− | = | + | is distributed according to the "chi-squared" law with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071040.png" /> degrees of freedom, then by virtue of their independence, the fraction |
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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/s09071041.png" /></td> </tr></table> | |
− | is subject to the Student distribution with | + | is subject to the Student distribution with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071042.png" /> degrees of freedom. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071043.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071044.png" /> be the solutions of the equations |
− | degrees of freedom. Let | ||
− | and | ||
− | be the solutions of the equations | ||
− | + | <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/s09071045.png" /></td> </tr></table> | |
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− | Then the statistics | + | Then the statistics <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071046.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071047.png" /> are the lower and upper bounds of the confidence set for the unknown mathematical expectation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071048.png" /> of the normal law <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071049.png" />, and the confidence coefficient of this confidence set is equal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/s/s090/s090710/s09071050.png" />, i.e. |
− | 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. | ||
− | + | <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/s09071051.png" /></td> </tr></table> | |
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The Student distribution was first used by W.S. Gosset (pseudonym Student). | The Student distribution was first used by W.S. Gosset (pseudonym Student). |
Revision as of 14:53, 7 June 2020
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=48882