DES - Demonstration of the First Property of the Transition Function Matrix

Arguments

DES - Definition of ‘Transition Function Matrix for CTHMC’

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Given the Matrix H(t) where $p_{i,j}(t)$ is defined as:

$$p_{i,j}(t)\triangleq P(X(s+t)=j\mid X(s)=i)$$

Then:

$$p_{i,j}(0)=P(X(s)=j\mid X(s)=i)=\{\begin{array}{cc}1& ifi=j\\ 0& ifi\ne j\end{array}$$

What is the probability that the state is $j$ when the state is $i$ : $0$ What is the probability that the state is $j$ when the state is $j$ : $1$