91Ó°ÊÓ

Consider a pure death process where \(\mu_{n}=n \mu\) for \(n=1,2, \ldots\), i.e., \(P\\{X(t+h)\) \(=j \mid X(t)=k\\}=0\) for \(j>k\) and \(t\) and \(k\) positive. Assume an initial population of size \(i\). Find \(P_{n}(t)=P\\{X(t)=n\\}, E X(t)\), and \(V\) ar \(X(t)\).

Consider a Poisson process of parameter \(\lambda\). Given that \(n\) events happen in time \(t\), find the density function of the time of the occurrence of the \(r\) th event \((r

Consider a pure birth process having infinitesimal parameters \(\lambda_{n}=\lambda n^{2}\), where \(\lambda>0\) is fixed. Given that at time 0 there is a single particle, determine $$ P_{\infty}(t)=1-\sum_{k=1}^{\infty} P_{k}(t) $$