Consider a linear birth process with state space \(\mathbf{Z}=\{0,1,2, \ldots\}\) and transition rates \(\lambda_{j}=j \lambda, j=0,1, \ldots\)

(1) Given \(X(0)=1\), determine the distribution function of the random time point \(T_{3}\) at which the process enters state 3 .

(2) Given \(X(0)=1\), determine the mean value of the random time point \(T_{n}\) at which the process enters state \(n, n>1\).