| 释义 |
Student's t-DistributionA Distribution published by William Gosset in 1908. His employer, Guinness Breweries, required him to publish under apseudonym, so he chose ``Student.'' Given  independent measurements  , let
 | (1) |
 is the population Mean,  is the sample Mean, and  is the Estimator for population Standard Deviation (i.e., the Sample Variance) defined by
 | (2) |
 -distribution is defined as the distribution of the random variable which is (very loosely) the ``best''that we can do not knowing  . If  ,  and the distribution becomes the Normal Distribution. As increases, Student's -distribution approaches the Normal Distribution.
Student's -distribution is arrived at by transforming to Student's z-Distribution with
 | (3) |
 | (4) |
where
 | (7) |
 ,  is the Gamma Function, is the Beta Function, and  is the Regularized Beta Function defined by
 | (8) |
The Mean, Variance, Skewness, and Kurtosis of Student's -distribution are
Beyer (1987, p. 514) 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. A partial table is givenbelow for small  and several common confidence intervals. | 80% | 90% | 95% | 99% | | 1 | 3.08 | 6.31 | 12.71 | 63.66 | | 2 | 1.89 | 2.92 | 4.30 | 9.92 | | 3 | 1.64 | 2.35 | 3.18 | 5.84 | | 4 | 1.53 | 2.13 | 2.78 | 4.60 | | 5 | 1.48 | 2.01 | 2.57 | 4.03 | | 10 | 1.37 | 1.81 | 2.23 | 4.14 | | 30 | 1.31 | 1.70 | 2.04 | 2.75 | | 100 | 1.29 | 1.66 | 1.98 | 2.63 |  | 1.28 | 1.65 | 1.96 | 2.58 |
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:
 | (13) |
 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
 | (14) |
The noncentral Student's -distribution is given by
where is the Gamma Function,  is a Confluent Hypergeometric Function,and  is an associated Laguerre Polynomial. See also Paired t-Test, Student's z-Distribution
References
Abramowitz, M. and Stegun, C. 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, p. 536, 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. 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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