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{{About|the mathematics of Student's ''t''-distribution|its uses in statistics|Student's t-test}}
In [[number theory]] '''Euler's criterion''' is a formula for determining whether an [[integer]] is a [[quadratic residue]] [[modular arithmetic|modulo]] a [[prime number|prime]]. Precisely,
{{More footnotes|date=February 2010}}
{{DISPLAYTITLE:Student's ''t''-distribution}}
{{Probability distribution |
  name      =Student's ''t''|
  type      =density|
  pdf_image  =[[Image:student t pdf.svg|325px]]|
  cdf_image  =[[Image:student t cdf.svg|325px]]|
  parameters =ν > 0 [[degrees of freedom (statistics)|degrees of freedom]] ([[Real number|real]])|
  support    =''x'' ∈ (−∞; +∞)|
  pdf        =<math>\textstyle\frac{\Gamma \left(\frac{\nu+1}{2} \right)} {\sqrt{\nu\pi}\,\Gamma \left(\frac{\nu}{2} \right)} \left(1+\frac{x^2}{\nu} \right)^{-\frac{\nu+1}{2}}\!</math>|
<!--For "<math>":
  --  Split formulas by "\begin{matrix}" with "\\[0.5em]" split --
  --  as 0.5em interline spacing; end with "\end{matrix}".      --
  --  Fractions have 2 brace-pairs. A centered dot is "\cdot". -->
  cdf        =<math>\begin{matrix}
    \frac{1}{2} + x \Gamma \left( \frac{\nu+1}{2} \right)  \times\\[0.5em]
    \frac{\,_2F_1 \left ( \frac{1}{2},\frac{\nu+1}{2};\frac{3}{2};
          -\frac{x^2}{\nu} \right)}
    {\sqrt{\pi\nu}\,\Gamma \left(\frac{\nu}{2}\right)}
    \end{matrix}</math><br/
    >where <sub>2</sub>''F''<sub>1</sub> is the [[hypergeometric function]]|
  mean      =0 for ν > 1, otherwise [[indeterminate form|undefined]]|
  median    =0|
  mode      =0|
  variance  =<math>\textstyle\frac{\nu}{\nu-2}</math> for ν > 2, ∞ for 1 < ν ≤ 2, otherwise [[indeterminate form|undefined]]|
  skewness  =0 for ν > 3, otherwise [[indeterminate form|undefined]]|
  kurtosis  =<math>\textstyle\frac{6}{\nu-4}</math> for ν > 4, ∞ for 2 < ν ≤ 4, otherwise [[indeterminate form|undefined]]|
  entropy    =<math>\begin{matrix}
        \frac{\nu+1}{2}\left[
            \psi \left(\frac{1+\nu}{2} \right)
              - \psi \left(\frac{\nu}{2} \right)
        \right] \\[0.5em]
+ \log{\left[\sqrt{\nu}B \left(\frac{\nu}{2},\frac{1}{2} \right)\right]}
\end{matrix}</math>
* ψ: [[digamma function]],
* ''B'': [[beta function]]|
|
  mgf        = undefined|
  char      =<math>\textstyle\frac{K_{\nu/2} \left(\sqrt{\nu}|t|\right)
                    \cdot \left(\sqrt{\nu}|t| \right)^{\nu/2}}
                    {\Gamma(\nu/2)2^{\nu/2-1}}</math> for ν > 0
* ''K''<sub>ν</sub>(''x''): [[Bessel function|Modified Bessel function of the second kind]]<ref>Hurst, Simon. ''[http://wwwmaths.anu.edu.au/research.reports/srr/95/044/ The Characteristic Function of the Student-t Distribution]'', Financial Mathematics Research Report No. FMRR006-95, Statistics Research Report No. SRR044-95{{dead link|date=April 2013}}</ref>
}}
In [[probability]] and [[statistics]], '''Student's  ''t''-distribution''' (or simply the '''''t''-distribution''') is a family of continuous [[probability distribution]]s that arise when estimating the [[expected value|mean]] of a [[normal distribution|normally distributed]] [[Statistical population|population]] in situations where the [[sample size]] is small and population [[standard deviation]] is unknown. It plays a role in a number of widely used statistical analyses, including the [[Student's t-test|Student's ''t''-test]] for assessing the [[statistical significance]] of the difference between two sample [[mean]]s, the construction of [[confidence interval]]s for the difference between two population means, and in linear [[regression analysis]]. The Student's ''t''-distribution also arises in the [[Bayesian analysis]] of data from a normal family.


If we take a sample of ''n = ν+1'' observations from a normal distribution (the black curve on the figure on the right of this page, representing a very large ''ν''), compute the sample mean and plot it, and repeat this process infinitely many times (for the same ''n''), we get the probability density function for that ''n'', as shown in the image on the right.
Let ''p'' be an [[odd number|odd]] prime and ''a'' an integer [[coprime]] to ''p''. Then<ref>Gauss, DA, Art. 106</ref>


If we also compute the [[Sample variance#Population variance and sample variance|sample variance]] for these ''n'' observations, then the ''t''-distribution (for ''n-1'') can be defined as the distribution of the location of the true mean, relative to the sample mean and divided by the sample standard deviation, after multiplying by the normalizing term <math>\sqrt{n}</math>, where ''n'' is the sample size. In this way, the ''t''-distribution can be used to estimate how likely it is that the true mean lies in any given range.
:<math>
a^{\tfrac{p-1}{2}} \equiv
\begin{cases}
\;\;\,1\pmod{p}& \text{ if there is an integer }x \text{ such that }a\equiv x^2 \pmod{p}\\
    -1\pmod{p}& \text{ if there is no such integer.}
\end{cases}
</math>


The ''t''-distribution is symmetric and bell-shaped, like the [[normal distribution]], but has heavier tails, meaning that it is more prone to producing values that fall far from its mean. This makes it useful for understanding the statistical behavior of certain types of ratios of random quantities, in which variation in the denominator is amplified and may produce outlying values when the denominator of the ratio falls close to zero.  The Student's ''t''-distribution is a special case of the [[generalised hyperbolic distribution]].
Euler's criterion can be concisely reformulated using the [[Legendre symbol]]:<ref>Hardy & Wright, thm. 83</ref>
:<math>
\left(\frac{a}{p}\right) \equiv a^{(p-1)/2} \pmod p.
</math>


==History and etymology==
The criterion first appeared in a 1748 paper by [[Leonhard Euler|Euler]].<ref>Lemmermeyer, p. 4 cites two papers, E134 and E262 in the Euler Archive</ref>
In statistics, the ''t''-distribution was first derived as a [[posterior distribution]] in 1876 by [[Friedrich Robert Helmert|Helmert]]<ref name=HFR1/><ref name=HFR2/><ref name=HFR3/> and [[Jacob Lüroth|Lüroth]].<ref name=L1876/><ref>{{Cite journal|first1=J.|last1=Pfanzagl| first2=O.|last2=Sheynin | title=A forerunner of the ''t''-distribution  (Studies in the history of probability and statistics XLIV) | year=1996 | journal=Biometrika | volume=83 | issue=4 |pages=891–898 | doi=10.1093/biomet/83.4.891 <!-- abstract=The t-distribution first occurred in a paper by Lüroth (1876) on the classical theory of errors in connection with a Bayesian result --> |mr=1766040}}</ref><ref>{{Cite journal| doi=10.1007/BF00374700 | last=Sheynin | first=O. | year=1995 | title=Helmert's work in the theory of errors | journal=Arch. Hist. Ex. Sci. | volume=49 | pages=73–104}}</ref>


In the English-language literature it takes its name from [[William Sealy Gosset]]'s 1908 paper in [[Biometrika]] under the pseudonym "Student".<ref>{{Cite journal|author="Student" <nowiki>[</nowiki>[[William Sealy Gosset]]<nowiki>]</nowiki>|date=March 1908 |title=The probable error of a mean |journal=[[Biometrika]] |volume=6 |issue=1 |pages=1–25 |url=http://www.york.ac.uk/depts/maths/histstat/student.pdf |doi=10.1093/biomet/6.1.1}}</ref><ref>"Student" (William Sealy Gosset), original Biometrika paper as a [http://www.atmos.washington.edu/~robwood/teaching/451/student_in_biometrika_vol6_no1.pdf scan]</ref> Gosset worked at the [[St. James's Gate Brewery|Guinness Brewery]] in [[Dublin, Ireland]], and was interested in the problems of small samples, for example of the chemical properties of barley where sample sizes might be as low as 3. One version of the origin of the pseudonym is that Gosset's employer forbade members of its staff from publishing scientific papers, so he had to hide his identity. Another version is that Guinness did not want their competitors to know that they were using the ''t''-test to test the quality of raw material.<ref>Mortimer, Robert G. (2005)  ''Mathematics for Physical Chemistry'', Academic Press. 3 edition. ISBN 0-12-508347-5 (page 326)</ref>
==Proof==


Gosset's paper refers to the distribution as the "frequency distribution of standard deviations of samples drawn from a normal population". It became well-known through the work of [[Ronald A. Fisher]], who called the distribution "Student's distribution" and referred to the value as ''t''.<ref name="Fisher 1925 90–104">{{Cite journal|last=Fisher |first=R. A. |authorlink=Ronald Fisher |year=1925 |title=Applications of "Student's" distribution |journal=Metron |volume=5 |pages=90–104 |url=http://www.sothis.ro/user/content/4ef6e90670749a86-student_distribution_1925.pdf}}</ref><ref>Walpole, Ronald; Myers, Raymond; Myers, Sharon; Ye, Keying. (2002) ''Probability and Statistics for Engineers and Scientists''. Pearson Education, 7th edition, pg. 237 ISBN 81-7758-404-9</ref>
The proof uses fact that the residue classes modulo a prime number are a [[finite field|field]]. See the article [[Characteristic_(algebra)#Case_of_fields|prime field]] for more details. The fact that there are (''p'' − 1)/2 quadratic residues and the same number of nonresidues (mod ''p'') is proved in the article [[quadratic residue]].


==Definition==
[[Fermat's little theorem]] says that
 
:<math>
===Probability density function===
a^{p-1}\equiv 1 \pmod p.
Student's '''''t''-distribution''' has the [[probability density function]] given by
</math>
 
This can be written as
:<math>f(t) = \frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} \left(1+\frac{t^2}{\nu} \right)^{-\frac{\nu+1}{2}},\!</math>
:<math>
 
(a^{\tfrac{p-1}{2}}-1)(a^{\tfrac{p-1}{2}}+1)\equiv 0 \pmod p.
where <math>\nu</math> is the number of ''[[degrees of freedom (statistics)|degrees of freedom]]'' and <math>\Gamma</math> is the [[gamma function]]. This may also be written as
</math>  
 
Since the integers mod ''p'' form
:<math>f(t) = \frac{1}{\sqrt{\nu}\, B \left (\frac{1}{2}, \frac{\nu}{2}\right )} \left(1+\frac{t^2}{\nu} \right)^{-\frac{\nu+1}{2}}\!,</math>
a field, one or the other of these factors must be congruent to zero.
 
where ''B'' is the [[Beta function]].
 
For <math>\nu</math> even,
 
: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} =  \frac{(\nu -1)(\nu -3)\cdots 5 \cdot 3} {2\sqrt{\nu}(\nu -2)(\nu -4)\cdots 4 \cdot 2\,}. </math>
For <math>\nu</math> odd,
 
: <math>\frac{\Gamma(\frac{\nu+1}{2})} {\sqrt{\nu\pi}\,\Gamma(\frac{\nu}{2})} =  \frac{(\nu -1)(\nu -3)\cdots 4 \cdot 2} {\pi \sqrt{\nu}(\nu -2)(\nu -4)\cdots 5 \cdot 3\,}.\!</math>
 
The probability density function is [[Symmetric distribution|symmetric]], and its overall shape resembles the bell shape of a [[normal distribution|normally distributed]] variable with mean 0 and variance 1, except that it is a bit lower and wider. As the number of degrees of freedom grows, the ''t''-distribution approaches the normal distribution with mean 0 and variance 1.
 
The following images show the density of the ''t''-distribution for increasing values of <math>\nu</math>.  The normal distribution is shown as a blue line for comparison.  Note that the ''t''-distribution (red line) becomes closer to the normal distribution as <math>\nu</math> increases.
 
{| align="center"
|+ Density of the ''t''-distribution (red) for 1, 2, 3, 5, 10, and 30 degrees of freedom compared to the standard normal distribution (blue).<br>Previous plots shown in green.
|-
| [[Image:T distribution 1df enhanced.svg|thumb|center|240x240px|alt=1df|1 degree of freedom]]
| [[Image:T distribution 2df enhanced.svg|thumb|center|240x240px|alt=2df|2 degrees of freedom]]
| [[Image:T distribution 3df enhanced.svg|thumb|center|240x240px|alt=3df|3 degrees of freedom]]
|-
| [[Image:T distribution 5df enhanced.svg|thumb|center|240x240px|alt=5df|5 degrees of freedom]]
| [[Image:T distribution 10df enhanced.svg|thumb|center|240x240px|alt=10df|10 degrees of freedom]]
| [[Image:T distribution 30df enhanced.svg|thumb|center|240x240px|alt=30df|30 degrees of freedom]]
|}
 
===Cumulative distribution function===
The [[cumulative distribution function]] can be written in terms of ''I'', the regularized
[[incomplete beta function]]. For ''t''&nbsp;>&nbsp;0,<ref name=JKB/>
 
:<math>F(t) = \int_{-\infty}^t f(u)\,du = 1- \tfrac{1}{2} I_{x(t)}\left(\tfrac{\nu}{2}, \tfrac{1}{2}\right),</math>
 
with
 
:<math>x(t) = \frac{\nu}{{t^2+\nu}}.</math>
 
Other values would be obtained by symmetry. An alternative formula, valid for ''t''<sup>2</sup> < ν, is<ref name=JKB/>
 
:<math>\int_{-\infty}^t f(u)\,du =\tfrac{1}{2} + t\frac{\Gamma \left( \tfrac{1}{2}(\nu+1) \right)} {\sqrt{\pi\nu}\,\Gamma \left(\tfrac{\nu}{2}\right)}  {}_2F_1 \left ( \tfrac{1}{2},\tfrac{1}{2}(\nu+1); \tfrac{3}{2};  -\tfrac{t^2}{\nu} \right)</math>
 
where <sub>2</sub>''F''<sub>1</sub> is a particular case of the [[hypergeometric function]].
 
===Special cases===
Certain values of ν give an especially simple form.
 
*ν = 1
 
:Distribution function:
 
::<math>F(x) = \tfrac{1}{2} + \tfrac{1}{\pi}\arctan(x).</math>
 
:Density function:
 
::<math>f(x) =  \frac{1}{\pi (1+x^2)}.</math>
 
:See [[Cauchy distribution]]
 
*ν = 2
:Distribution function:
 
::<math>F(x) = \tfrac{1}{2}+\frac{x}{2\sqrt{2+x^2}}.</math>
 
:Density function:
 
::<math>f(x) = \frac{1}{\left(2+x^2\right)^{\frac{3}{2}}}.</math>
 
*ν = 3
 
:Density function:
 
::<math>f(x) = \frac{6\sqrt{3}}{\pi\left(3+x^2\right)^2}.</math>
 
*ν = ∞
 
:Density function:


::<math>f(x) = \frac{1}{\sqrt{2\pi}} e^{-\frac{x^2}{2}}.</math>
Now if ''a'' is a quadratic residue, ''a'' &equiv; ''x''<sup>2</sup>,
 
:<math>
:See [[Normal distribution]]
a^{\tfrac{p-1}{2}}\equiv{x^2}^{\tfrac{p-1}{2}}\equiv x^{p-1}\equiv1\pmod p.
 
==How the ''t''-distribution arises==
 
===Sampling distribution===
Let ''x''<sub>1</sub>, ..., ''x''<sub>''n''</sub> be the numbers observed in a sample from a continuously distributed population with expected value μ. The sample mean and [[sample variance]] are given by:
 
<math>
\begin{align}
\bar{x} &= \frac{x_1+\cdots+x_n}{n} \\
s^2 &= \frac{1}{n-1}\sum_{i=1}^n (x_i - \bar{x})^2
\end{align}
</math>
</math>
So every quadratic residue (mod ''p'') makes the first factor zero.


The resulting ''t-value'' is
[[Lagrange's theorem (number theory)|Lagrange's theorem]] says that there can be no more than (''p''&nbsp;−&nbsp;1)/2 values of ''a'' that make the first factor zero. But it is known that there are (''p''&nbsp;−&nbsp;1)/2 distinct quadratic residues (mod ''p''). Therefore they are precisely the residue classes that make the first factor zero. The other (''p''&nbsp;−&nbsp;1)/2 residue classes, the nonresidues, must be the ones making the second factor zero. This is Euler's criterion.


: <math> t = \frac{\bar{x} - \mu}{s/\sqrt{n}}. </math>
==Examples==
'''Example 1: Finding primes for which ''a'' is a residue'''


The ''t''-distribution with ''n'' − 1 degrees of freedom is the [[sampling distribution]] of the ''t''-value when the samples consist of [[independent identically distributed]] observations from a [[normal distribution|normally distributed]] population. Thus for inference purposes ''t'' is a useful "[[pivotal quantity]]" in the case when the mean and variance (μ, σ<sup>2</sup>) are unknown population parameters, in the sense that the ''t''-value has then a probability distribution that depends on neither μ nor σ<sup>2</sup>.
Let ''a'' = 17. For which primes ''p'' is 17 a quadratic residue?


<span id="Bayesiantdistribution">
We can test prime ''p'''s manually given the formula above.


===Bayesian inference===
In one case, testing ''p'' = 3, we have 17<sup>(3 − 1)/2</sup> = 17<sup>1</sup> ≡ 2 ≡ −1 (mod 3), therefore 17 is not a quadratic residue modulo 3.
In Bayesian statistics, a (scaled, shifted) ''t''-distribution arises as the marginal distribution of the unknown mean of a normal distribution, when the dependence on an unknown variance has been marginalised out:<ref>A. Gelman ''et al'' (1995), ''Bayesian Data Analysis'', Chapman & Hall. ISBN 0-412-03991-5. p. 68</ref>


:<math>\begin{align}
In another case, testing ''p'' = 13, we have 17<sup>(13 − 1)/2</sup> = 17<sup>6</sup> ≡ 1 (mod 13), therefore 17 is a quadratic residue modulo 13.  As confirmation, note that 17 ≡ 4 (mod 13), and 2<sup>2</sup> = 4.
p(\mu|D, I) = & \int p(\mu,\sigma^2|D, I) \; d \sigma^2 \\
= & \int p(\mu|D, \sigma^2, I) \; p(\sigma^2|D, I) \; d \sigma^2
\end{align}</math>


where ''D'' stands for the data {''x''<sub>i</sub>} and ''I'' represents any other information that may have been used to create the model. The distribution is thus the [[compound distribution|compounding]] of the conditional distribution of μ given the data and σ<sup>2</sup> with the marginal distribution of σ<sup>2</sup> given the data.<br>
We can do these calculations faster by using various modular arithmetic and Legendre symbol properties.
<!--
:<math>p(\mu|D, \sigma^2, I) \propto \int p(D|\mu, \sigma^2, I) \; p(\mu|\sigma^2, I)</math>
and
:<math>\begin{align}p(\sigma^2|D, I) = & \int p(\mu, \sigma^2|D, I) \; d\mu \\
\propto & \int p(D |\sigma^2, \mu, I) \; p(\mu|\sigma^2, I) \; p(\sigma^2 | I)\end{align}</math>
-->
With ''n'' data points, if [[Jeffreys prior|uninformative]] location and scale priors <math>\scriptstyle{p(\mu|\sigma^2, I) = \mbox{const}}</math> and <math>\scriptstyle{p(\sigma^2|I)\; \propto \;1/\sigma^2}</math> can be taken for μ and σ<sup>2</sup>, then [[Bayes' theorem]] gives


:<math>\begin{align}
If we keep calculating the values, we find:
p(\mu|D, \sigma^2, I) \sim & N(\bar{x}, \sigma^2/n) \\
:(17/''p'') = +1 for ''p'' = {13, 19, ...} (17 is a quadratic residue modulo these values)
p(\sigma^2 | D, I) \sim & \operatorname{Scale-inv-}\chi^2(\nu, s^2)
\end{align}</math>


a Normal distribution and a [[scaled inverse chi-squared distribution]] respectively, where ν = ''n'' − 1 and
:(17/''p'') = −1 for ''p'' = {3, 5, 7, 11, 23, ...} (17 is not a quadratic residue modulo these values).


:<math>s^2 = \sum \frac{(x_i - \bar{x})^2}{n-1}</math>.
'''Example 2: Finding residues given a prime modulus ''p'' '''


The marginalisation integral thus becomes
Which numbers are squares modulo 17 (quadratic residues modulo 17)?


:<math>\begin{align}
We can manually calculate:
p(\mu|D, I) &\propto \int_0^{\infty} \frac{1}{\sqrt{\sigma^2}} \exp \left(-\frac{1}{2\sigma^2} n(\mu - \bar{x})^2\right) \;\cdot\; \sigma^{-\nu-2}\exp(-\nu s^2/2 \sigma^2) \; d\sigma^2 \\
: 1<sup>2</sup> = 1
&\propto \int_0^{\infty} \sigma^{-\nu-3} \exp \left(-\frac{1}{2 \sigma^2} \left(n(\mu - \bar{x})^2 + \nu s^2\right) \right)  \; d\sigma^2
: 2<sup>2</sup> = 4
\end{align}</math>
: 3<sup>2</sup> = 9
: 4<sup>2</sup> = 16
: 5<sup>2</sup> = 25 ≡ 8 (mod 17)
: 6<sup>2</sup> = 36 ≡ 2 (mod 17)
: 7<sup>2</sup> = 49 ≡ 15 (mod 17)
: 8<sup>2</sup> = 64 ≡ 13 (mod 17).


This can be evaluated by substituting <math>\scriptstyle{z = A / 2\sigma^2}</math>, where <math>\scriptstyle{A = n(\mu - \bar{x})^2 + \nu s^2}</math>, giving
So the set of the quadratic residues modulo 17 is {1,2,4,8,9,13,15,16}.  Note that we did not need to calculate squares for the values 9 through 16, as they are all negatives of the previously squared values (e.g. 9 ≡ −8 (mod 17), so 9<sup>2</sup> ≡ (−8)<sup>2</sup> = 64 ≡ 13 (mod 17)).
:<math>dz = -\frac{A}{2 \sigma^4} d \sigma^2,</math>
so


:<math>p(\mu|D, I) \propto \; A^{-\frac{\nu + 1}{2}} \int_0^\infty z^{(\nu-1)/2} \exp(-z) dz</math>
We can find quadratic residues or verify them using the above formula.  To test if 2 is a quadratic residue modulo 17, we calculate 2<sup>(17 − 1)/2</sup> = 2<sup>8</sup> ≡ 1 (mod 17), so it is a quadratic residue.  To test if 3 is a quadratic residue modulo 17, we calculate 3<sup>(17 − 1)/2</sup> = 3<sup>8</sup> ≡ 16 ≡ −1 (mod 17), so it is not a quadratic residue.


But the ''z'' integral is now a standard [[Gamma integral]], which evaluates to a constant, leaving
Euler's criterion is related to the [[Quadratic reciprocity|Law of quadratic reciprocity]] and is used in a definition of [[Euler–Jacobi pseudoprime]]s.


:<math>\begin{align}p(\mu|D, I) \propto & \; A^{-\frac{\nu + 1}{2}} \\
==See also==
\propto & \left( 1 + \frac{n(\mu - \bar{x})^2}{\nu s^2} \right)^{-\frac{\nu + 1}{2}} \end{align}</math>


This is a form of the ''t'' distribution with an explicit scaling and shifting that will be explored in more detail in a further section below.  It can be related to the standardised ''t'' distribution by the substitution
==Notes==


:<math>t = \frac{\mu - \bar{x}}{s / \sqrt{n}}</math>
{{reflist}}


The derivation above has been presented for the case of uninformative priors for μ and σ<sup>2</sup>; but it will be apparent that any priors which lead to a Normal distribution being compounded with a scaled inverse chi-squared distribution will lead to a ''t'' distribution with scaling and shifting for ''P''(μ|''D'',''I''), although the scaling parameter corresponding to ''s''<sup>2</sup>/''n'' above will then be influenced both by the prior information and the data, rather than just by the data as above.
==References==


</span>
The ''[[Disquisitiones Arithmeticae]]'' has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.


==Characterization==


===As the distribution of a test statistic===
*{{citation
Student's ''t''-distribution with ν degrees of freedom can be defined as the distribution of the [[random variable]] ''T'' with <ref name=JKB>Johnson, N.L., Kotz, S., Balakrishnan, N. (1995)
  | last1 = Gauss  | first1 = Carl Friedrich
''Continuous Univariate Distributions, Volume 2,'' 2nd Edition. Wiley, ISBN 0-471-58494-0 (Chapter 28)</ref><ref name=Hogg>Hogg & Craig (1978, Sections 4.4 and 4.8.)</ref>
  | last2 = Clarke | first2 = Arthur A. (translator into English) 
  | title = Disquisitiones Arithemeticae (Second, corrected edition)
  | publisher = [[Springer Science+Business Media|Springer]]
  | location = New York
  | date = 1986
  | isbn = 0-387-96254-9}}


:<math> T=\frac{Z}{\sqrt{V/\nu}} = Z \sqrt{\frac{\nu}{V}} ,</math>
*{{citation
  | last1 = Gauss  | first1 = Carl Friedrich
  | last2 = Maser | first2 = H. (translator into German) 
  | title = Untersuchungen uber hohere Arithmetik (Disquisitiones Arithemeticae & other papers on number theory) (Second edition)
  | publisher = Chelsea
  | location = New York
  | date = 1965
  | isbn = 0-8284-0191-8}}


where


* ''Z'' is [[normal distribution|normally distributed]] with [[expected value]] 0 and variance 1;
*{{citation
* ''V'' has a [[chi-squared distribution]] with ν [[Degrees of freedom (statistics)|degrees of freedom]];
  | last1 = Hardy  | first1 = G. H.
* ''Z'' and ''V'' are [[statistical independence|independent]].
  | last2 = Wright | first2 = E. M.
  | title = An Introduction to the Theory of Numbers (Fifth edition)
  | publisher = [[Oxford University Press]]
  | location = Oxford
  | date = 1980
  | isbn = 978-0-19-853171-5}}


A different distribution is defined as that of the random variable defined, for a given constant&nbsp;μ, by
*{{citation
:<math>(Z+\mu)\sqrt{\frac{\nu}{V}}.</math>
  | last1 = Lemmermeyer  | first1 = Franz
This random variable has a [[noncentral t-distribution|noncentral ''t''-distribution]] with [[noncentrality parameter]] μ. This distribution is important in studies of the [[statistical power|power]] of Student's ''t''-test.
  | title = Reciprocity Laws: from Euler to Eisenstein
  | publisher = [[Springer Science+Business Media|Springer]]
  | location = Berlin
  | date = 2000
  | isbn = 3-540-66957-4}}


====Derivation====
==External links==
Suppose ''X''<sub>1</sub>, ..., ''X''<sub>''n''</sub> are [[statistical independence|independent]] random variables that are normally distributed with expected value μ and [[variance]] σ<sup>2</sup>. Let
 
:<math>\overline{X}_n = \frac{1}{n}(X_1+\cdots+X_n)</math>
 
be the sample mean, and
 
:<math>S_n^{\;2} = \frac{1}{n-1}\sum_{i=1}^n\left(X_i-\overline{X}_n\right)^2</math>
 
be an unbiased estimate of the variance from the sample.  It can be shown that the random variable
 
: <math>V = (n-1)\frac{S_n^2}{\sigma^2} </math>
 
has a [[chi-squared distribution]] with ''v=n−1'' degrees of freedom (by [[Cochran's theorem]]).<ref>{{cite journal|last=Cochran|first=W. G.|authorlink=William Gemmell Cochran|title=The distribution of quadratic forms in a normal system, with applications to the analysis of covariance|journal=[[Mathematical Proceedings of the Cambridge Philosophical Society]]|date=April 1934|volume=30|issue=2|pages=178–191|doi=10.1017/S0305004100016595|bibcode = 1934PCPS...30..178C }}</ref>  It is readily shown that the quantity
 
:<math>Z = \left(\overline{X}_n-\mu\right)\frac{\sqrt{n}}{\sigma}</math>
 
is normally distributed with mean 0 and variance 1, since the sample mean <math>\overline{X}_n</math> is normally distributed with mean μ and variance σ<sup>2</sup>/''n''.  Moreover, it is possible to show that these two random variables (the normally distributed one '''''Z''''' and the chi-squared-distributed one '''''V''''') are independent.  Consequently{{clarify|date=November 2012}} the [[pivotal quantity]],
 
:<math>T \equiv \frac{Z}{\sqrt{V/v}} = \left(\overline{X}_n-\mu\right)\frac{\sqrt{n}}{S_n},</math>
 
which differs from ''Z'' in that the exact standard deviation σ is replaced by the random variable ''S''<sub>''n''</sub>, has a Student's ''t''-distribution as defined above. Notice that the unknown population variance σ<sup>2</sup> does not appear in ''T'', since it was in both the numerator and the denominator, so it canceled. Gosset intuitively obtained the [[probability density function]] stated above, with ν equal to ''n'' − 1, and Fisher proved it in 1925.<ref name="Fisher 1925 90–104"/>
 
The distribution of the test statistic, ''T'', depends on ν, but not μ or σ; the lack of dependence on μ and σ is what makes the ''t''-distribution important in both theory and practice.
 
===As a maximum entropy distribution===
Student's ''t''-distribution is the [[maximum entropy probability distribution]] for a random variate ''X'' for which <math>E(\ln(\nu+X^2))</math> is fixed.<ref>{{cite journal |last1=Park |first1=Sung Y. |last2=Bera |first2=Anil K. |year=2009 |title=Maximum entropy autoregressive conditional heteroskedasticity model |journal=Journal of Econometrics |volume= |issue= |pages=219–230 |publisher=Elsevier |doi= |url=http://www.wise.xmu.edu.cn/Master/Download/..%5C..%5CUploadFiles%5Cpaper-masterdownload%5C2009519932327055475115776.pdf |accessdate=2011-06-02 }}</ref>
 
==Properties==
 
===Moments===
The [[raw moment]]s of the ''t''-distribution are
 
:<math>E(T^k)=\begin{cases}
0 & k \text{ odd},\quad 0<k< \nu\\
\frac{1}{\sqrt{\pi}\Gamma\left(\frac{\nu}{2}\right)}\left[\Gamma\left(\frac{k+1}{2}\right)\Gamma\left(\frac{\nu-k}{2}\right)\nu^{\frac{k}{2}}\right] & k \text{ even}, \quad 0<k< \nu\\
\end{cases}</math>
 
Moments of order ν or higher do not exist.<ref>See, for example, page 56 of Casella and Berger, ''Statistical Inference'', 1990 Duxbury.</ref>
 
The term for 0 < ''k'' < ν, ''k'' even, may be simplified using the properties of the [[gamma function]] to
 
:<math>E(T^k)= \nu^{\frac{k}{2}} \, \prod_{i=1}^{\frac{k}{2}} \frac{2i-1}{\nu - 2i} \qquad k\text{ even},\quad 0<k<\nu. </math>
 
For a ''t''-distribution with ν degrees of freedom, the [[expected value]] is 0, and its [[variance]] is ν/(ν − 2) if ν > 2. The [[skewness]] is 0 if ν > 3 and the [[excess kurtosis]] is 6/(ν − 4) if ν > 4.


===Relation to F distribution===
*[http://www.math.dartmouth.edu/~euler/index.html The Euler Archive]
*<math>Y \sim \mathrm{F}(\nu_1 = 1, \nu_2 = \nu)</math> has an [[F-distribution|''F''-distribution]] if ''Y'' = ''X''<sup>2</sup> and ''X'' ~ t(ν) has a Student's ''t''-distribution.
 
===Monte Carlo sampling===
There are various approaches to constructing random samples from the Student's ''t''-distribution. The matter depends on whether the samples are required on a stand-alone basis, or are to be constructed by application of a [[quantile function]] to [[uniform]] samples; e.g., in the multi-dimensional applications basis of [[Copula (statistics)|copula-dependency]].{{citation needed|date=July 2011}} In the case of stand-alone sampling, an extension of the [[Box–Muller method]] and its [[Box–Muller transform#Polar form|polar form]] is easily deployed.<ref name=Bailey>{{Cite journal
| last1 = Bailey | first1 = R. W.
| title = Polar Generation of Random Variates with the ''t''-Distribution
| journal = Mathematics of Computation
| volume = 62
| issue = 206
| pages = 779–781
| doi = 10.2307/2153537
| year = 1994
| pmid = 
| pmc =
}}</ref> It has the merit that it applies equally well to all real positive [[degrees of freedom (statistics)|degrees of freedom]], ν, while many other candidate methods fail if ν is close to zero.<ref name=Bailey/>
 
===Integral of Student's probability density function and ''p''-value===
The function ''A''(''t''|ν) is the integral of Student's probability density function, ''f''(''t'') between −''t'' and ''t'', for ''t'' ≥ 0.  It thus gives the probability that a value of ''t'' less than that calculated from observed data would occur by chance.  Therefore, the function ''A''(''t''|ν) can be used when testing whether the difference between the means of two sets of data is statistically significant, by calculating the corresponding value of ''t'' and the probability of its occurrence if the two sets of data were drawn from the same population.  This is used in a variety of situations, particularly in [[T test|''t''-tests]].  For the statistic ''t'', with ν degrees of freedom, ''A''(''t''|ν) is the probability that ''t'' would be less than the observed value if the two means were the same (provided that the smaller mean is subtracted from the larger, so that ''t'' ≥ 0).  It can be easily calculated from the [[cumulative distribution function]] ''F''<sub>ν</sub>(''t'') of the ''t''-distribution:
 
:<math>A(t|\nu) = F_\nu(t) - F_\nu(-t) = 1 - I_{\frac{\nu}{\nu +t^2}}\left(\frac{\nu}{2},\frac{1}{2}\right),</math>
 
where ''I''<sub>''x''</sub> is the regularized [[Beta function#Incomplete beta function|incomplete beta function]] (''a'',&nbsp;''b'').
 
For statistical hypothesis testing this function is used to construct the [[p-value|''p''-value]].
 
=={{anchor|Three-parameter version|Non-standardized}}Non-standardized Student's ''t''-distribution==
 
===In terms of scaling parameter σ, or σ<sup>2</sup>===
Student's t distribution can be generalized to a three parameter [[location-scale family]], introducing a [[location parameter]] μ and a [[scale parameter]] σ, through the relation
:<math>X = \mu + \sigma T</math>
The resulting '''non-standardized Student's ''t''-distribution''' has a density defined by<ref name="Jackman" >{{Cite book |last= Jackman |first= Simon |title= Bayesian Analysis for the Social Sciences |publisher= Wiley |year=2009 |page=507}}</ref>
 
:<math>p(x|\nu,\mu,\sigma) = \frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})\sqrt{\pi\nu}\sigma} \left(1+\frac{1}{\nu}\left(\frac{x-\mu}{\sigma}\right)^2\right)^{-\frac{\nu+1}{2}} </math>
 
Here, σ does ''not'' correspond to a [[standard deviation]]: it is not the standard deviation of the scaled ''t'' distribution, which may not even exist; nor is it the standard deviation of the underlying [[normal distribution]], which is unknown. σ simply sets the overall scaling of the distribution.  In the Bayesian derivation of the marginal distribution of an unknown Normal mean μ above, σ as used here corresponds to the quantity <math>\scriptstyle{s/\sqrt{n}}</math>, where
 
:<math>s^2 = \sum \frac{(x_i - \bar{x})^2}{n-1}</math>.
 
Equivalently, the distribution can be written in terms of σ<sup>2</sup>, the square of this scale parameter:
 
:<math>p(x|\nu,\mu,\sigma^2) = \frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})\sqrt{\pi\nu\sigma^2}} \left(1+\frac{1}{\nu}\frac{(x-\mu)^2}{\sigma^2}\right)^{-\frac{\nu+1}{2}} </math>
 
Other properties of this version of the distribution are:<ref name="Jackman" />
 
:<math>\begin{align}
\operatorname{E}(X) &= \mu \quad \quad \quad \text{for }\,\nu > 1 ,\\
\text{var}(X) &= \sigma^2\frac{\nu}{\nu-2}\, \quad \text{for }\,\nu > 2 ,\\
\text{mode}(X) &= \mu.
\end{align} </math>
 
This distribution results from [[compound distribution|compounding]] a [[Gaussian distribution]] ([[normal distribution]]) with [[mean]] μ and unknown [[variance]], with an [[inverse gamma distribution]] placed over the variance with parameters ''a'' = ν/2 and <math>b = \nu\sigma^2/2</math>. In other words, the [[random variable]] ''X'' is assumed to have a Gaussian distribution with an unknown variance distributed as inverse gamma, and then the variance is [[marginalized out]] (integrated out). The reason for the usefulness of this characterization is that the inverse gamma distribution is the [[conjugate prior]] distribution of the variance of a Gaussian distribution. As a result, the non-standardized Student's ''t''-distribution arises naturally in many [[Bayesian inference]] problems.  See below.
 
Equivalently, this distribution results from compounding a Gaussian distribution with a [[scaled-inverse-chi-squared distribution]] with parameters ν and σ<sup>2</sup>. The scaled-inverse-chi-squared distribution is exactly the same distribution as the inverse gamma distribution, but with a different parameterization, i.e. ν = ''a''/2, σ<sup>2</sup> = ''b''/''a''.
 
===In terms of inverse scaling parameter λ===
An alternative [[parameterization]] in terms of an inverse scaling parameter λ (analogous to the way [[precision (statistics)|precision]] is the reciprocal of variance), defined by the relation λ = σ<sup>−2</sup>.  Then the density is defined by<ref name="Bishop2006">{{Cite book |last= Bishop |first= C.M. |title= Pattern recognition and machine learning |publisher= [[Springer Science+Business Media|Springer]] |year=2006}}</ref>
 
:<math>p(x|\nu,\mu,\lambda) = \frac{\Gamma(\frac{\nu + 1}{2})}{\Gamma(\frac{\nu}{2})} \left(\frac{\lambda}{\pi\nu}\right)^{\frac{1}{2}} \left(1+\frac{\lambda(x-\mu)^2}{\nu}\right)^{-\frac{\nu+1}{2}}.</math>
 
Other properties of this version of the distribution are:<ref name="Bishop2006" />
 
:<math> \begin{align}
\operatorname{E}(X) &= \mu \quad \quad \quad \text{for }\,\nu > 1 ,\\
\text{var}(X) &= \frac{1}{\lambda}\frac{\nu}{\nu-2}\, \quad \text{for }\,\nu > 2 ,\\
\text{mode}(X) &= \mu.
\end{align} </math>
 
This distribution results from [[compound distribution|compounding]] a [[Gaussian distribution]] with [[mean]] μ and unknown [[precision (statistics)|precision]] (the reciprocal of the [[variance]]), with a [[gamma distribution]] placed over the precision with parameters ''a'' = ν/2 and ''b'' = ν/(2λ).  In other words, the random variable ''X'' is assumed to have a [[normal distribution]] with an unknown precision distributed as gamma, and then this is marginalized over the gamma distribution.
 
==Related distributions==
 
===Noncentral ''t''-distribution===
The [[noncentral t-distribution|noncentral ''t''-distribution]] is a different way of generalizing the ''t''-distribution to include a location parameter.  Unlike the nonstandardized ''t''-distributions, the noncentral distributions are not symmetric (the median is not the same as the mode).
 
===Discrete Student's ''t''-distribution===
The '''discrete Student's ''t''-distribution''' is defined by its [[probability mass function]] at ''r'' being proportional to<ref>Ord, J.K. (1972) ''Families of Frequency Distributions'', Griffin. ISBN 0-85264-137-0 (Table 5.1)</ref>
:<math> \prod_{j=1}^k \frac{1}{(r+j+a)^2+b^2}  \quad \quad r=\ldots, -1, 0, 1, \ldots  .</math>
Here ''a'', ''b'', and ''k'' are parameters.
This distribution arises from the construction of a system of discrete distributions similar to that of the [[Pearson distribution]]s for continuous distributions.<ref>Ord, J.K. (1972) ''Families of Frequency Distributions'', Griffin. ISBN 0-85264-137-0 (Chapter 5)</ref>
 
==Uses==
 
===In frequentist statistical inference===
Student's ''t''-distribution arises in a variety of statistical estimation problems where the goal is to estimate an unknown parameter, such as a mean value, in a setting where the data are observed with additive [[errors and residuals in statistics|errors]].  If (as in nearly all practical statistical work) the population [[standard deviation]] of these errors is unknown and has to be estimated from the data, the ''t''-distribution is often used to account for the extra uncertainty that results from this estimation.  In most such problems, if the standard deviation of the errors were known, a [[normal distribution]] would be used instead of the ''t''-distribution.
 
[[Confidence interval]]s and [[hypothesis test]]s are two statistical procedures in which the [[quantile]]s of the sampling distribution of a particular statistic (e.g. the [[standard score]]) are required.  In any situation where this statistic is a [[linear function]] of the [[data]], divided by the usual estimate of the standard deviation, the resulting quantity can be rescaled and centered to follow Student's ''t''-distribution.  Statistical analyses involving means, weighted means, and regression coefficients all lead to statistics having this form.
 
Quite often, textbook problems will treat the population standard deviation as if it were known and thereby avoid the need to use the Student's ''t''-distribution. These problems are generally of two kinds: (1) those in which the sample size is so large that one may treat a data-based estimate of the [[variance]] as if it were certain, and (2) those that illustrate mathematical reasoning, in which the problem of estimating the standard deviation is temporarily ignored because that is not the point that the author or instructor is then explaining.
 
====Hypothesis testing====
A number of statistics can be shown to have ''t''-distributions for samples of moderate size under [[null hypothesis|null hypotheses]] that are of interest, so that the ''t''-distribution forms the basis for significance tests.  For example, the distribution of [[Spearman's rank correlation coefficient]] ''ρ'', in the null case (zero correlation) is well approximated by the ''t'' distribution for sample sizes above about 20 .{{citation needed|date=November 2010}}
 
====Confidence intervals====
Suppose the number ''A'' is so chosen that
 
:<math>\Pr(-A < T < A)=0.9,</math>
 
when ''T'' has a ''t''-distribution with ''n'' − 1 degrees of freedom.  By symmetry, this is the same as saying that ''A'' satisfies
 
:<math>\Pr(T < A) = 0.95,</math>
 
so ''A'' is the "95th percentile" of this probability distribution, or <math> A=t_{(0.05,n-1)}</math>.  Then
 
:<math>\Pr \left (-A < \frac{\overline{X}_n - \mu}{\frac{S_n}{\sqrt{n}}} < A \right)=0.9,</math>
 
and this is equivalent to
 
:<math>\Pr\left(\overline{X}_n - A \frac{S_n}{\sqrt{n}} < \mu < \overline{X}_n + A\frac{S_n}{\sqrt{n}}\right) = 0.9.</math>
 
Therefore the interval whose endpoints are
 
:<math>\overline{X}_n\pm A\frac{S_n}{\sqrt{n}}</math>
 
is a 90% [[confidence interval]] for μ.  Therefore, if we find the mean of a set of observations that we can reasonably expect to have a normal distribution, we can use the ''t''-distribution to examine whether the confidence limits on that mean include some theoretically predicted value – such as the value predicted on a [[null hypothesis]].
 
It is this result that is used in the [[Student's t-test|Student's ''t''-test]]s: since the difference between the means of samples from two normal distributions is itself distributed normally, the ''t''-distribution can be used to examine whether that difference can reasonably be supposed to be zero.
 
If the data are normally distributed, the one-sided (1 − ''a'')-upper confidence limit (UCL) of the mean, can be calculated using the following equation:
 
:<math>\mathrm{UCL}_{1-a} = \overline{X}_n + t_{a,n-1}\frac{S_n}{\sqrt{n}}.</math>
 
The resulting UCL will be the greatest average value that will occur for a given confidence interval and population size. In other words, <math>\overline{X}_n</math> being the mean of the set of observations, the probability that the mean of the distribution is inferior to UCL<sub>1−''a''</sub> is equal to the confidence level 1 − ''a''.
 
====Prediction intervals====
The ''t''-distribution can be used to construct a [[prediction interval]] for an unobserved sample from a normal distribution with unknown mean and variance.
 
===In Bayesian statistics===
The Student's ''t''-distribution, especially in its three-parameter (location-scale) version, arises frequently in [[Bayesian statistics]] as a result of its connection with the [[normal distribution]].  Whenever the [[variance]] of a normally distributed [[random variable]] is unknown and a [[conjugate prior]] placed over it that follows an [[inverse gamma distribution]], the resulting [[marginal distribution]] of the variable will follow a Student's ''t''-distribution.  Equivalent constructions with the same results involve a conjugate [[scaled-inverse-chi-squared distribution]] over the variance, or a conjugate [[gamma distribution]] over the [[precision (statistics)|precision]].  If an [[improper prior]] proportional to σ<sup>−2</sup> is placed over the variance, the ''t''-distribution also arises. This is the case regardless of whether the mean of the normally distributed variable is known, is unknown distributed according to a [[conjugate prior|conjugate]] normally distributed prior, or is unknown distributed according to an improper constant prior.
 
Related situations that also produce a ''t''-distribution are:
*The [[marginal distribution|marginal]] [[posterior distribution]] of the unknown mean of a normally distributed variable, with unknown prior mean and variance following the above model.
*The [[prior predictive distribution]] and [[posterior predictive distribution]] of a new normally distributed data point when a series of [[independent identically distributed]] normally distributed data points have been observed, with prior mean and variance as in the above model.
 
===Robust parametric modeling===
The ''t''-distribution is often used as an alternative to the normal distribution as a model for data.<ref>{{Cite journal| last=Lange | first=Kenneth L. | coauthors=Little, Roderick J.A.; Taylor, Jeremy M.G. | journal=JASA | title=Robust statistical modeling using the ''t''-distribution | year=1989 | volume=84 | issue=408 | pages=881–896 | jstor=2290063}}</ref> It is frequently the case that real data have heavier tails than the normal distribution allows for. The classical approach was to identify outliers and exclude or downweight them in some way. However, it is not always easy to identify outliers (especially in [[curse of dimensionality|high dimensions]]), and the ''t''-distribution is a natural choice of model for such data and provides a parametric approach to [[robust statistics]].
 
Lange et al. explored the use of the ''t''-distribution for robust modeling of heavy tailed data in a variety of contexts. A Bayesian account can be found in Gelman et al. The degrees of freedom parameter controls the kurtosis of the distribution and is correlated with the scale parameter. The likelihood can have multiple local maxima and, as such, it is often necessary to fix the degrees of freedom at a fairly low value and estimate the other parameters taking this as given. Some authors report that values between 3 and 9 are often good choices. Venables and Ripley suggest that a value of 5 is often a good choice.
 
==Table of selected values==
Most statistical textbooks list ''t'' distribution tables. Nowadays, the better way to a fully precise critical ''t'' value or a cumulative probability is the statistical function implemented in spreadsheets (Office Excel, OpenOffice Calc, etc.), or an interactive calculating web page. The relevant spreadsheet functions are TDIST and TINV, while online calculating pages save troubles like positions of parameters or names of functions. For example, a [[MediaWiki]] page supported by [[R (programming language)|R]] extension can easily give the interactive result of critical values or cumulative probability, even for noncentral ''t''-distribution.
 
The following table lists a few selected values for ''t''-distributions with ν degrees of freedom for a range of ''one-sided'' or ''two-sided'' critical regions. For an example of how to read this table, take the fourth row, which begins with 4; that means ν, the number of degrees of freedom, is 4 (and if we are dealing, as above, with ''n'' values with a fixed sum, ''n'' = 5). Take the fifth entry, in the column headed 95% for ''one-sided'' (90% for ''two-sided''). The value of that entry is "2.132". Then the probability that ''T'' is less than 2.132 is 95% or Pr(−∞ < ''T'' < 2.132) = 0.95; or mean that Pr(−2.132 < ''T'' < 2.132) = 0.9.
 
This can be calculated by the symmetry of the distribution,
 
:Pr(''T''&nbsp;<&nbsp;−2.132)&nbsp;=&nbsp;1&nbsp;−&nbsp;Pr(''T'' > −2.132) = 1 − 0.95 = 0.05,
 
and so
 
: Pr(−2.132&nbsp;<&nbsp;''T''&nbsp;<&nbsp;2.132) = 1&nbsp;−&nbsp;2(0.05) = 0.9.
 
'''Note''' that the last row also gives critical points: a ''t''-distribution with infinitely many degrees of freedom is a normal distribution. (See [[#Related distributions|Related distributions]] above).
 
The first column is the number of degrees of freedom.
 
{| class="wikitable"
|-
! ''One Sided''
! '''75%'''
! '''80%'''
! '''85%'''
! '''90%'''
! '''95%'''
! '''97.5%'''
! '''99%'''
! '''99.5%'''
! '''99.75%'''
! '''99.9%'''
! '''99.95%'''
|-
! ''Two Sided''
! '''50%'''
! '''60%'''
! '''70%'''
! '''80%'''
! '''90%'''
! '''95%'''
! '''98%'''
! '''99%'''
! '''99.5%'''
! '''99.8%'''
! '''99.9%'''
|-
!'''1'''
|1.000
|1.376
|1.963
|3.078
|6.314
|12.71
|31.82
|63.66
|127.3
|318.3
|636.6
|-
!'''2'''
|0.816
|1.061
|1.386
|1.886
|2.920
|4.303
|6.965
|9.925
|14.09
|22.33
|31.60
|-
!'''3'''
|0.765
|0.978
|1.250
|1.638
|2.353
|3.182
|4.541
|5.841
|7.453
|10.21
|12.92
|-
!'''4'''
|0.741
|0.941
|1.190
|1.533
|2.132
|2.776
|3.747
|4.604
|5.598
|7.173
|8.610
|-
!'''5'''
|0.727
|0.920
|1.156
|1.476
|2.015
|2.571
|3.365
|4.032
|4.773
|5.893
|6.869
|-
!'''6'''
|0.718
|0.906
|1.134
|1.440
|1.943
|2.447
|3.143
|3.707
|4.317
|5.208
|5.959
|-
!'''7'''
|0.711
|0.896
|1.119
|1.415
|1.895
|2.365
|2.998
|3.499
|4.029
|4.785
|5.408
|-
!'''8'''
|0.706
|0.889
|1.108
|1.397
|1.860
|2.306
|2.896
|3.355
|3.833
|4.501
|5.041
|-
!'''9'''
|0.703
|0.883
|1.100
|1.383
|1.833
|2.262
|2.821
|3.250
|3.690
|4.297
|4.781
|-
!'''10'''
|0.700
|0.879
|1.093
|1.372
|1.812
|2.228
|2.764
|3.169
|3.581
|4.144
|4.587
|-
!'''11'''
|0.697
|0.876
|1.088
|1.363
|1.796
|2.201
|2.718
|3.106
|3.497
|4.025
|4.437
|-
!'''12'''
|0.695
|0.873
|1.083
|1.356
|1.782
|2.179
|2.681
|3.055
|3.428
|3.930
|4.318
|-
!'''13'''
|0.694
|0.870
|1.079
|1.350
|1.771
|2.160
|2.650
|3.012
|3.372
|3.852
|4.221
|-
!'''14'''
|0.692
|0.868
|1.076
|1.345
|1.761
|2.145
|2.624
|2.977
|3.326
|3.787
|4.140
|-
!'''15'''
|0.691
|0.866
|1.074
|1.341
|1.753
|2.131
|2.602
|2.947
|3.286
|3.733
|4.073
|-
!'''16'''
|0.690
|0.865
|1.071
|1.337
|1.746
|2.120
|2.583
|2.921
|3.252
|3.686
|4.015
|-
!'''17'''
|0.689
|0.863
|1.069
|1.333
|1.740
|2.110
|2.567
|2.898
|3.222
|3.646
|3.965
|-
!'''18'''
|0.688
|0.862
|1.067
|1.330
|1.734
|2.101
|2.552
|2.878
|3.197
|3.610
|3.922
|-
!'''19'''
|0.688
|0.861
|1.066
|1.328
|1.729
|2.093
|2.539
|2.861
|3.174
|3.579
|3.883
|-
!'''20'''
|0.687
|0.860
|1.064
|1.325
|1.725
|2.086
|2.528
|2.845
|3.153
|3.552
|3.850
|-
!'''21'''
|0.686
|0.859
|1.063
|1.323
|1.721
|2.080
|2.518
|2.831
|3.135
|3.527
|3.819
|-
!'''22'''
|0.686
|0.858
|1.061
|1.321
|1.717
|2.074
|2.508
|2.819
|3.119
|3.505
|3.792
|-
!'''23'''
|0.685
|0.858
|1.060
|1.319
|1.714
|2.069
|2.500
|2.807
|3.104
|3.485
|3.767
|-
!'''24'''
|0.685
|0.857
|1.059
|1.318
|1.711
|2.064
|2.492
|2.797
|3.091
|3.467
|3.745
|-
!'''25'''
|0.684
|0.856
|1.058
|1.316
|1.708
|2.060
|2.485
|2.787
|3.078
|3.450
|3.725
|-
!'''26'''
|0.684
|0.856
|1.058
|1.315
|1.706
|2.056
|2.479
|2.779
|3.067
|3.435
|3.707
|-
!'''27'''
|0.684
|0.855
|1.057
|1.314
|1.703
|2.052
|2.473
|2.771
|3.057
|3.421
|3.690
|-
!'''28'''
|0.683
|0.855
|1.056
|1.313
|1.701
|2.048
|2.467
|2.763
|3.047
|3.408
|3.674
|-
!'''29'''
|0.683
|0.854
|1.055
|1.311
|1.699
|2.045
|2.462
|2.756
|3.038
|3.396
|3.659
|-
!'''30'''
|0.683
|0.854
|1.055
|1.310
|1.697
|2.042
|2.457
|2.750
|3.030
|3.385
|3.646
|-
!'''40'''
|0.681
|0.851
|1.050
|1.303
|1.684
|2.021
|2.423
|2.704
|2.971
|3.307
|3.551
|-
!'''50'''
|0.679
|0.849
|1.047
|1.299
|1.676
|2.009
|2.403
|2.678
|2.937
|3.261
|3.496
|-
!'''60'''
|0.679
|0.848
|1.045
|1.296
|1.671
|2.000
|2.390
|2.660
|2.915
|3.232
|3.460
|-
!'''80'''
|0.678
|0.846
|1.043
|1.292
|1.664
|1.990
|2.374
|2.639
|2.887
|3.195
|3.416
|-
!'''100'''
|0.677
|0.845
|1.042
|1.290
|1.660
|1.984
|2.364
|2.626
|2.871
|3.174
|3.390
|-
!'''120'''
|0.677
|0.845
|1.041
|1.289
|1.658
|1.980
|2.358
|2.617
|2.860
|3.160
|3.373
|-
!'''<math>\infty</math>'''
|0.674
|0.842
|1.036
|1.282
|1.645
|1.960
|2.326
|2.576
|2.807
|3.090
|3.291
|}
 
The number at the beginning of each row in the table above is ν which has been defined above as ''n'' − 1.  The percentage along the top is 100%(1 − α).  The numbers in the main body of the table are ''t''<sub>α, ν</sub>.  If a quantity ''T'' is distributed as a Student's t distribution with ν degrees of freedom, then there is a probability 1 − α that ''T'' will be less than ''t''<sub>α, ν</sub>. (Calculated as for a one-tailed or one-sided test, as opposed to a [[two-tailed test]].)
 
For example, given a sample with a sample variance 2 and sample mean of 10, taken from a sample set of 11 (10 degrees of freedom), using the formula
 
:<math>\overline{X}_n\pm A\frac{S_n}{\sqrt{n}}.</math>
 
We can determine that at 90% confidence, we have a true mean lying below
 
:<math>10+1.37218 \frac{\sqrt{2}}{\sqrt{11}}=10.58510.</math>
 
(In other words, on average, 90% of the times that an upper threshold is calculated by this method, this upper threshold exceeds the true mean.)  And, still at 90% confidence, we have a true mean lying over
 
:<math>10-1.37218 \frac{\sqrt{2}}{\sqrt{11}}=9.41490.</math>
 
(In other words, on average, 90% of the times that a lower threshold is calculated by this method, this lower threshold lies below the true mean.)  So that at 80% confidence (calculated from 1&nbsp;−&nbsp;2&nbsp;×&nbsp;(1&nbsp;−&nbsp;90%) = 80%), we have a true mean lying within the interval
 
:<math>\left(10-1.37218 \frac{\sqrt{2}}{\sqrt{11}}, 10+1.37218 \frac{\sqrt{2}}{\sqrt{11}}\right) = \left(9.41490, 10.58510\right). </math>
 
This is generally expressed in interval notation, e.g., for this case, at 80% confidence the true mean is within the interval [9.41490,&nbsp;10.58510].
 
(In other words, on average, 80% of the times that upper and lower thresholds are calculated by this method, the true mean is both below the upper threshold and above the lower threshold. This is not the same thing as saying that there is an 80% probability that the true mean lies between a particular pair of upper and lower thresholds that have been calculated by this method—see [[confidence interval]] and [[prosecutor's fallacy]].)
 
For information on the inverse cumulative distribution function see ''[[quantile function]]''.
 
==See also==
{{Portal|Statistics}}
{{Colbegin}}
* [[Chi-squared distribution]]
* [[F-distribution|''F''-distribution]]
* [[Gamma distribution]]
* [[Hotelling's T-squared distribution|Hotelling's ''T''-squared distribution]]
* [[Multivariate Student distribution]]
* [[t-statistic|''t''-statistic]]
* [[Wilks' lambda distribution]]
* [[Wishart distribution]]
{{Colend}}
 
==Notes==
{{Reflist|30em|refs=
 
<ref name=HFR1>Helmert, F. R. (1875). "Über die Bestimmung des wahrscheinlichen Fehlers aus einer endlichen Anzahl wahrer Beobachtungsfehler". ''Z. Math. Phys.'', 20, 300–3.</ref>
<ref name=HFR2>Helmert, F. R. (1876a). "Über die Wahrscheinlichkeit der Potenzsummen der Beobachtungsfehler und uber einige damit in Zusammenhang stehende Fragen". ''Z. Math. Phys.'', 21, 192–218.</ref>
<ref name=HFR3>Helmert, F. R. (1876b). "Die Genauigkeit der Formel von Peters zur Berechnung des wahrscheinlichen Beobachtungsfehlers directer Beobachtungen gleicher Genauigkeit", ''Astron. Nachr.'', 88, 113–32.</ref>
<ref name=L1876>{{Cite journal| doi=10.1002/asna.18760871402 | last=Lüroth | first=J | year=1876 | title=Vergleichung von zwei Werten des wahrscheinlichen Fehlers | journal=Astron. Nachr. | volume=87 | pages=209–20| issue=14 | bibcode=1876AN.....87..209L}}</ref>
}}
 
==References==
*{{Cite journal
| last1 = Senn | first1 = S.
| last2 = Richardson | first2 = W.
| title = The first ''t''-test
| journal = [[Statistics in Medicine (journal)|Statistics in Medicine]]
| volume = 13
| issue = 8
| pages = 785–803
| year = 1994
| pmid = 8047737 |doi=10.1002/sim.4780130802
}}
* [[Robert V. Hogg|Hogg, R.V.]]; Craig, A.T. (1978). ''Introduction to Mathematical Statistics''. New York: Macmillan.
*Venables, W.N.; B.D. Ripley, B.D. (2002)''Modern Applied Statistics with S'', Fourth Edition, Springer
*{{Cite book
  | last = Gelman
  | first = Andrew
  | coauthors = John B. Carlin, Hal S. Stern, Donald B. Rubin
  | title = Bayesian Data Analysis (Second Edition)
  | publisher = CRC/Chapman & Hall
  | year = 2003
  | isbn = 1-58488-388-X
  | url = http://www.stat.columbia.edu/~gelman/book/
}}
 
==External links==
*{{springer|title=Student distribution|id=p/s090710}}
*[http://calculus-calculator.com/statistics/students-t-distribution-calculator.html Calculator for the pdf, cdf and critical values of the Student's t-distribution]
*[http://jeff560.tripod.com/s.html Earliest Known Uses of Some of the Words of Mathematics (S)] ''(Remarks on the history of the term "Student's distribution")''


{{ProbDistributions|continuous-infinite}}
{{DEFAULTSORT:Euler's Criterion}}
{{Common univariate probability distributions|state=collapsed}}
[[Category:Modular arithmetic]]
{{Statistics|state=collapsed}}
[[Category:Articles containing proofs]]
[[Category:Quadratic residue]]
[[Category:Theorems about prime numbers]]


{{DEFAULTSORT:Student's T-Distribution}}
[[ca:Criteri d'Euler]]
[[Category:Continuous distributions]]
[[es:Criterio de Euler]]
[[Category:Special functions]]
[[fr:Critère d'Euler]]
[[Category:Normal distribution]]
[[it:Criterio di Eulero]]
[[Category:Probability distributions with non-finite variance]]
[[he:מבחן אוילר]]
[[Category:Infinitely divisible probability distributions]]
[[pl:Kryterium Eulera]]
[[Category:Probability distributions]]
[[sv:Eulers kriterium]]
[[zh:欧拉准则]]

Revision as of 02:28, 10 August 2014

In number theory Euler's criterion is a formula for determining whether an integer is a quadratic residue modulo a prime. Precisely,

Let p be an odd prime and a an integer coprime to p. Then[1]

ap−12≡{1(modp) if there is an integer x such that a≡x2(modp)−1(modp) if there is no such integer.

Euler's criterion can be concisely reformulated using the Legendre symbol:[2]

(ap)≡a(p−1)/2(modp).

The criterion first appeared in a 1748 paper by Euler.[3]

Proof

The proof uses fact that the residue classes modulo a prime number are a field. See the article prime field for more details. The fact that there are (p − 1)/2 quadratic residues and the same number of nonresidues (mod p) is proved in the article quadratic residue.

Fermat's little theorem says that

ap−1≡1(modp).

This can be written as

(ap−12−1)(ap−12+1)≡0(modp).

Since the integers mod p form a field, one or the other of these factors must be congruent to zero.

Now if a is a quadratic residue, a ≡ x2,

ap−12≡x2p−12≡xp−1≡1(modp).

So every quadratic residue (mod p) makes the first factor zero.

Lagrange's theorem says that there can be no more than (p − 1)/2 values of a that make the first factor zero. But it is known that there are (p − 1)/2 distinct quadratic residues (mod p). Therefore they are precisely the residue classes that make the first factor zero. The other (p − 1)/2 residue classes, the nonresidues, must be the ones making the second factor zero. This is Euler's criterion.

Examples

Example 1: Finding primes for which a is a residue

Let a = 17. For which primes p is 17 a quadratic residue?

We can test prime p's manually given the formula above.

In one case, testing p = 3, we have 17(3 − 1)/2 = 171 ≡ 2 ≡ −1 (mod 3), therefore 17 is not a quadratic residue modulo 3.

In another case, testing p = 13, we have 17(13 − 1)/2 = 176 ≡ 1 (mod 13), therefore 17 is a quadratic residue modulo 13. As confirmation, note that 17 ≡ 4 (mod 13), and 22 = 4.

We can do these calculations faster by using various modular arithmetic and Legendre symbol properties.

If we keep calculating the values, we find:

(17/p) = +1 for p = {13, 19, ...} (17 is a quadratic residue modulo these values)
(17/p) = −1 for p = {3, 5, 7, 11, 23, ...} (17 is not a quadratic residue modulo these values).

Example 2: Finding residues given a prime modulus p

Which numbers are squares modulo 17 (quadratic residues modulo 17)?

We can manually calculate:

12 = 1
22 = 4
32 = 9
42 = 16
52 = 25 ≡ 8 (mod 17)
62 = 36 ≡ 2 (mod 17)
72 = 49 ≡ 15 (mod 17)
82 = 64 ≡ 13 (mod 17).

So the set of the quadratic residues modulo 17 is {1,2,4,8,9,13,15,16}. Note that we did not need to calculate squares for the values 9 through 16, as they are all negatives of the previously squared values (e.g. 9 ≡ −8 (mod 17), so 92 ≡ (−8)2 = 64 ≡ 13 (mod 17)).

We can find quadratic residues or verify them using the above formula. To test if 2 is a quadratic residue modulo 17, we calculate 2(17 − 1)/2 = 28 ≡ 1 (mod 17), so it is a quadratic residue. To test if 3 is a quadratic residue modulo 17, we calculate 3(17 − 1)/2 = 38 ≡ 16 ≡ −1 (mod 17), so it is not a quadratic residue.

Euler's criterion is related to the Law of quadratic reciprocity and is used in a definition of Euler–Jacobi pseudoprimes.

See also

Notes

43 year old Petroleum Engineer Harry from Deep River, usually spends time with hobbies and interests like renting movies, property developers in singapore new condominium and vehicle racing. Constantly enjoys going to destinations like Camino Real de Tierra Adentro.

References

The Disquisitiones Arithmeticae has been translated from Gauss's Ciceronian Latin into English and German. The German edition includes all of his papers on number theory: all the proofs of quadratic reciprocity, the determination of the sign of the Gauss sum, the investigations into biquadratic reciprocity, and unpublished notes.


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    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010


  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010
  • Many property agents need to declare for the PIC grant in Singapore. However, not all of them know find out how to do the correct process for getting this PIC scheme from the IRAS. There are a number of steps that you need to do before your software can be approved.

    Naturally, you will have to pay a safety deposit and that is usually one month rent for annually of the settlement. That is the place your good religion deposit will likely be taken into account and will kind part or all of your security deposit. Anticipate to have a proportionate amount deducted out of your deposit if something is discovered to be damaged if you move out. It's best to you'll want to test the inventory drawn up by the owner, which can detail all objects in the property and their condition. If you happen to fail to notice any harm not already mentioned within the inventory before transferring in, you danger having to pay for it yourself.

    In case you are in search of an actual estate or Singapore property agent on-line, you simply should belief your intuition. It's because you do not know which agent is nice and which agent will not be. Carry out research on several brokers by looking out the internet. As soon as if you end up positive that a selected agent is dependable and reliable, you can choose to utilize his partnerise in finding you a home in Singapore. Most of the time, a property agent is taken into account to be good if he or she locations the contact data on his website. This may mean that the agent does not mind you calling them and asking them any questions relating to new properties in singapore in Singapore. After chatting with them you too can see them in their office after taking an appointment.

    Have handed an trade examination i.e Widespread Examination for House Brokers (CEHA) or Actual Property Agency (REA) examination, or equal; Exclusive brokers are extra keen to share listing information thus making certain the widest doable coverage inside the real estate community via Multiple Listings and Networking. Accepting a severe provide is simpler since your agent is totally conscious of all advertising activity related with your property. This reduces your having to check with a number of agents for some other offers. Price control is easily achieved. Paint work in good restore-discuss with your Property Marketing consultant if main works are still to be done. Softening in residential property prices proceed, led by 2.8 per cent decline within the index for Remainder of Central Region

    Once you place down the one per cent choice price to carry down a non-public property, it's important to accept its situation as it is whenever you move in – faulty air-con, choked rest room and all. Get round this by asking your agent to incorporate a ultimate inspection clause within the possibility-to-buy letter. HDB flat patrons routinely take pleasure in this security net. "There's a ultimate inspection of the property two days before the completion of all HDB transactions. If the air-con is defective, you can request the seller to repair it," says Kelvin.

    15.6.1 As the agent is an intermediary, generally, as soon as the principal and third party are introduced right into a contractual relationship, the agent drops out of the image, subject to any problems with remuneration or indemnification that he could have against the principal, and extra exceptionally, against the third occasion. Generally, agents are entitled to be indemnified for all liabilities reasonably incurred within the execution of the brokers´ authority.

    To achieve the very best outcomes, you must be always updated on market situations, including past transaction information and reliable projections. You could review and examine comparable homes that are currently available in the market, especially these which have been sold or not bought up to now six months. You'll be able to see a pattern of such report by clicking here It's essential to defend yourself in opposition to unscrupulous patrons. They are often very skilled in using highly unethical and manipulative techniques to try and lure you into a lure. That you must also protect your self, your loved ones, and personal belongings as you'll be serving many strangers in your home. Sign a listing itemizing of all of the objects provided by the proprietor, together with their situation. HSR Prime Recruiter 2010

External links

ca:Criteri d'Euler es:Criterio de Euler fr:Critère d'Euler it:Criterio di Eulero he:מבחן אוילר pl:Kryterium Eulera sv:Eulers kriterium zh:欧拉准则

  1. ↑ Gauss, DA, Art. 106
  2. ↑ Hardy & Wright, thm. 83
  3. ↑ Lemmermeyer, p. 4 cites two papers, E134 and E262 in the Euler Archive
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