In this paper, thePascal-Beta and thePascal-Gamma distributions have been derived by compounding the Pascal distribution with the Beta distribution and the Gamma distribution. These two compound distributions include Pascal-Uniform, Geometric-Beta, Geometric-Uniform, Pascal-Exponential, Geometric-Gamma and Geometric-Exponential as special cases. The methods of computing the moments of these compound distributions have been given. The moment estimators of the parameters of the Pascal-Beta distribution have been derived explicitly and the necessary condition has been pointed out for the asymptotic normality of these estimators. The essential expressions have been derived to obtain the maximum likelihood estimators of the parameters of the Pascal-Beta distribution numerically. The moment estimators of the parameters of the Binomial-Beta distribution have been also derived explicitly. It has been pointed out that under certain conditions the Pascal-Beta distribution can be approximated by the Pascal distribution, the Geometric-Beta distribution can be approximated by the Geometric distribution, and the Binomial-Beta distribution can be approximated by the Binomial distribution. The paper also contains some discussions on the methods of obtaining the moment and maximum likelihood estimators of the parameters of the Pascal-Gamma distribution.