Student,s t-Distribution
المؤلف:
Abramowitz, M. and Stegun, I. A.
المصدر:
Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover
الجزء والصفحة:
...
14-4-2021
4060
Student's t-Distribution
A statistical distribution published by William Gosset in 1908. His employer, Guinness Breweries, required him to publish under a pseudonym, so he chose "Student." Given 
independent measurements 
, let
 |
(1)
|
where 
is the population mean, 
is the sample mean, and 
is the estimator for population standard deviation (i.e., the sample variance) defined by
 |
(2)
|
Student's 
-distribution is defined as the distribution of the random variable 
which is (very loosely) the "best" that we can do not knowing 
.
The Student's 
-distribution with 
degrees of freedom is implemented in the Wolfram Language as StudentTDistribution[n].
If 
, 
and the distribution becomes the normal distribution. As 
increases, Student's 
-distribution approaches the normal distribution.
Student's 
-distribution can be derived by transforming Student's z-distribution using
 |
(3)
|
and then defining
 |
(4)
|
The resulting probability and cumulative distribution functions are
where
 |
(10)
|
is the number of degrees of freedom, 
, 
is the gamma function, 
is the beta function, 
is a hypergeometric function, and 
is the regularized beta function defined by
 |
(11)
|
The mean, variance, skewness, and kurtosis excess of Student's 
-distribution are
The characteristic functions 
for the first few values of 
are
and so on, where 
is a modified Bessel function of the second kind.
The following table gives confidence intervals, i.e., values of 
such that the distribution function 
equals various probabilities for various small values of the numbers of degrees of freedom 
. Beyer (1987, p. 571) gives 60%, 70%, 90%, 95%, 97.5%, 99%, 99.5%, and 99.95% confidence intervals, and Goulden (1956) gives 50%, 90%, 95%, 98%, 99%, and 99.9% confidence intervals.
 |
90% |
95% |
97.5% |
99.5% |
| 1 |
3.07768 |
6.31375 |
12.7062 |
63.6567 |
| 2 |
1.88562 |
2.91999 |
4.30265 |
9.92484 |
| 3 |
1.63774 |
2.35336 |
3.18245 |
5.84091 |
| 4 |
1.53321 |
2.13185 |
2.77645 |
4.60409 |
| 5 |
1.47588 |
2.01505 |
2.57058 |
4.03214 |
| 10 |
1.37218 |
1.81246 |
2.22814 |
3.16927 |
| 30 |
1.31042 |
1.69726 |
2.04227 |
2.75000 |
| 100 |
1.29007 |
1.66023 |
1.98397 |
2.62589 |
 |
1.28156 |
1.64487 |
1.95999 |
2.57588 |
A multivariate form of the Student's 
-distribution with correlation matrix 
and 
degrees of freedom is implemented as MultivariateTDistribution[r, m] in the Wolfram Language package MultivariateStatistics` .
The so-called 
distribution is useful for testing if two observed distributions have the same mean. 
gives the probability that the difference in two observed means for a certain statistic 
with 
degrees of freedom would be smaller than the observed value purely by chance:
 |
(21)
|
Let 
be a normally distributed random variable with mean 0 and variance 
, let 
have a chi-squared distribution with 
degrees of freedom, and let 
and 
be independent. Then
 |
(22)
|
is distributed as Student's 
with 
degrees of freedom.
REFERENCES:
Abramowitz, M. and Stegun, I. A. (Eds.). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, pp. 948-949, 1972.
Beyer, W. H. CRC Standard Mathematical Tables, 28th ed. Boca Raton, FL: CRC Press, pp. 536 and 571, 1987.
Fisher, R. A. "Applications of 'Student's' Distribution." Metron 5, 3-17, 1925.
Fisher, R. A. "Expansion of 'Student's' Integral in Powers of 
." Metron 5, 22-32, 1925.
Fisher, R. A. Statistical Methods for Research Workers, 10th ed. Edinburgh: Oliver and Boyd, 1948.
Goulden, C. H. Table A-3 in Methods of Statistical Analysis, 2nd ed. New York: Wiley, p. 443, 1956.
Press, W. H.; Flannery, B. P.; Teukolsky, S. A.; and Vetterling, W. T. "Incomplete Beta Function, Student's Distribution, F-Distribution, Cumulative Binomial Distribution." §6.2 in Numerical Recipes in FORTRAN: The Art of Scientific Computing, 2nd ed. Cambridge, England: Cambridge University Press, pp. 219-223, 1992.
Shaw, W. "New Methods for Managing 'Student's' T-Distribution." Submitted to J. Comput. Finance. https://www.mth.kcl.ac.uk/~shaww/web_page/papers/Tdistribution06.pdf.
Spiegel, M. R. Theory and Problems of Probability and Statistics. New York: McGraw-Hill, pp. 116-117, 1992.
Student. "The Probable Error of a Mean." Biometrika 6, 1-25, 1908.
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