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Behrens–Fisher distribution : ウィキペディア英語版
Behrens–Fisher distribution

In statistics, the Behrens–Fisher distribution, named after Ronald Fisher and W. V. Behrens, is a parameterized family of probability distributions arising from the solution of the Behrens–Fisher problem proposed first by Behrens and several years later by Fisher. The Behrens–Fisher problem is that of statistical inference concerning the difference between the means of two normally distributed populations when the ratio of their variances is not known (and in particular, it is not known that their variances are equal).
== Definition ==

The Behrens–Fisher distribution is the distribution of a random variable of the form
: T_2 \cos\theta - T_1\sin\theta \,
where ''T''1 and ''T''2 are independent random variables each with a Student's t-distribution, with respective degrees of freedom ''ν''1 = ''n''1 − 1 and ''ν''2 = ''n''2 − 1, and ''θ'' is a constant. Thus the family of Behrens–Fisher distributions is parametrized by ''ν''1, ''ν''2, and ''θ''.

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