With the Rouse-Zimm differential equation of the spring-bead model, the distribution function ofN + 1 beads(x, y, z,t) [here x denotes x0, x1,..., xN, and similarly for y and z] is explicitly solved with the two different initial conditions: the Gaussian and delta distribution functions. We find that although the mean end-to-end distances obtained from the two initial conditions are the same, the expressions of the mean square end-to-end distances are different. We also obtain the expression for the mean and mean square end-to-end distances analytically from the Langevin equation with the delta initial distribution function. With this analytic expression, we show that the statistical quantities obtained from the Monte Carlo calculation are consistent with those obtained from the Rouse-Zimm differential equation if a suitable length is chosen for the time increment.