A basic doubt on the sojourn time of a CTMC

Question

By using the memoryless property, I find the CDF of sojourn time of a CTMC as follows :

$$F_X(t) = 1-e^{-F_X'(0)t}$$

I am slightly confused about the intuitive meaning of the term $F_X'(0)$. How this term represents the rate of leaving that state.

Answer 1

These are the basics of the exponential distributions, probably explained on the WP page.

If $T$ is exponential with parameter $a$, then $F_T(t)=1-\mathrm e^{-at}$ for every $t\geqslant0$ hence $a=F_T'(0^+)$ and $F_T(t)=1-\mathrm e^{-F_T'(0^+)t}$.

Furthermore, for every $t\geqslant0$, $P[t\leqslant T\leqslant t+s\mid T\geqslant t]=1-\mathrm e^{-as}\sim as$ when $s\to0$ hence, indeed, $a$ represents the rate of $T$ happening at $t$. Exponential distributions are characterized by the fact that this rate does not depend on $t\geqslant0$.