An Exact Expression for the Behrens-Fisher Distribution with Applications

An exact solution is given for the Behrens-Fisher distribution under the independent Normal model. The cumulative distribution function (cdf) and the probability density function (pdf) are expressed as convex combinations of non-central t-distributions cdfs and pdfs, respectively. It is then observed that if the means are equal the distribution depends only on the ratio of the two population variances and the sample sizes. When the means are not equal, the distribution depends on the difference of the two means, the ratio of the two population variances, and the two sample sizes. Some examples are given applying these results.