The graph of the exponential distribution is shown in Figure 1.

Figure 1: Graph of Markov Process for Exponential Distribution. (Figure by MIT OpenCourseWare.)

The transition equations are

\[ p(0,t + \delta t) = \mu \delta tp(1,t) + p(0,t) + o(\delta t) \tag 1 \]

\[ p(1,t + \delta t) = (1 - \mu \delta t) p (1,t) + (0)p (0,t) + o(\delta t) \tag 2 \]

or,

\[\frac{dp(0,t)}{dt}= \mu p(1,t) \tag 3 \]

\[\frac{dp(1,t)}{dt}= -\mu p(1,t) \tag 4 \]

Solve differential equations (3), (4) and we get

\[p(0,t) = 1 - e^{-\mu t}\tag 5\]

\[p(1,t) = e^{-\mu t}\tag 6\]

Function \(p(0,t)\) is actually the cumulative density function of the exponential distribution

\[F(t) = p(0,t)\tag 7\]

Then the density function of the exponential distribution is

\[f(t) = \frac{dF(t)}{dt}= \mu e^{- \mu t}\tag 8\]