In probability theory, the telegraph process is a memoryless continuous-time stochastic process that shows two distinct values. It models burst noise (also called popcorn noise or random telegraph signal). If the two possible values that a random variable can take are ${\displaystyle c_{1}}$ and ${\displaystyle c_{2}}$, then the process can be described by the following master equations:

and

where ${\displaystyle \lambda _{1}}$ is the transition rate for going from state ${\displaystyle c_{1}}$ to state ${\displaystyle c_{2}}$ and ${\displaystyle \lambda _{2}}$ is the transition rate for going from going from state ${\displaystyle c_{2}}$ to state ${\displaystyle c_{1}}$. The process is also known under the names Kac process (after mathematician Mark Kac), and dichotomous random process.

The master equation is compactly written in a matrix form by introducing a vector ${\displaystyle \mathbf {P} =[P(c_{1},t|x,t_{0}),P(c_{2},t|x,t_{0})]}$,

where

is the transition rate matrix. The formal solution is constructed from the initial condition ${\displaystyle \mathbf {P} (0)}$ (that defines that at ${\displaystyle t=t_{0}}$, the state is ${\displaystyle x}$) by

It can be shown that

where ${\displaystyle I}$ is the identity matrix and ${\displaystyle \lambda =(\lambda _{1}+\lambda _{2})/2}$ is the average transition rate. As ${\displaystyle t\rightarrow \infty }$, the solution approaches a stationary distribution ${\displaystyle \mathbf {P} (t\rightarrow \infty )=\mathbf {P} _{s}}$ given by