This paper deals with a generalization of type-1 Dirichlet density by incorporating partial sums of the component variables. We study various proportions, structural decompositions, connections to random volumes and p-parallelotopes. We will also look into the regression function of xk on x1,...,xk-1, Bayes’ estimates for the probabilities of a multinomial distribution by using this generalized Dirichlet model as the prior density are given. Other results illustrate the importance of the study of variable x1 in this model. It is found that the variable x1 in this model can be represented as the ratio of squares of volumes of two parallelotopes. Under certain conditions, x1 can be used to study the structural representations of the likelihood ratio criteria in MANOVA, MANCOVA etc.