Abstract: For odd degrees of freedom the characteristic function of a Student-t random variable is expressible in closed form. The characteristic function of an arbitrary linear combination of independent t-variables is then derived and the distribution function is obtained, itself expressible as a weighted sum of Student-t distribution functions. An easy method of obtaining the weights is demonstrated. If U1, U1, ..., Un are independent random variables and Xi = d + Ui, i=1,2,...,n are observable random variables, we investigate the choice of a1, a2, ...,an to maximize the power of tests of the form a1X1 + a2X2 + ... + anXn for testing Ho: d = 0 against H1: d > 0. Some general results and examples are given. Of particular interest is the case when Xi is a t-random variable. One application is in a two-stage sampling procedure to solve the Behrens-Fisher problem. The test statistic has the distribution of a weighted sum of t-random variables. It is shown how to choose the weights for maximum power. Dissertation Discovery Company and University of Florida are dedicated to making scholarly works more discoverable and accessible throughout the world. This dissertation, "On the Distribution of a Linear Combination of T-distributed Variables" by Glenn Alan Walker, was obtained from University of Florida and is being sold with permission from the author. A digital copy of this work may also be found in the university's institutional repository, IR@UF. The content of this dissertation has not been altered in any way. We have altered the formatting in order to facilitate the ease of printing and reading of the dissertation.