When being in states 0,1 , and 2 , a (pure) birth process \(\{X(t), t \geq 0\}\) with state space \(\mathbf{Z}=\{0,1,2, \ldots\}\) has the respective birth rates

\[\lambda_{0}=2, \lambda_{1}=3, \lambda_{2}=1\]

Given \(X(0)=0\), determine the time-dependent state probabilities \(p_{i}(t)=P(X(t)=i)\) for \(i=0,1,2\).