# An Exact Expression for the Behrens-Fisher Distribution with Applications

## Location

Room 1909

## Session Format

Paper Presentation

## Research Area Topic:

Natural & Physical Sciences - Mathematics

## Co-Presenters and Faculty Mentors or Advisors

Fengjiao Hu

## Abstract

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 infinite series 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 two population variances, and the two sample sizes. Methods are given for using the exact distribution to obtain an estimated confidence interval and an estimated *p*-value.

## Keywords

Comparing two population means, Confidence interval, p-value, Welsh’s approximation

## Presentation Type and Release Option

Presentation (Open Access)

## Start Date

4-24-2015 1:30 PM

## End Date

4-24-2015 2:30 PM

## Recommended Citation

Champ, Charles W., "An Exact Expression for the Behrens-Fisher Distribution with Applications" (2015). *GS4 Georgia Southern Student Scholars Symposium*. 71.

https://digitalcommons.georgiasouthern.edu/research_symposium/2015/2015/71

An Exact Expression for the Behrens-Fisher Distribution with Applications

Room 1909

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 infinite series 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 two population variances, and the two sample sizes. Methods are given for using the exact distribution to obtain an estimated confidence interval and an estimated *p*-value.