# Telegraph process

In probability theory, the telegraph process is a memoryless continuous-time stochastic process that shows two distinct values.

If these are called a and b, the process can be described by the following master equations:

$\partial_t P(a, t|x, t_0)=-\lambda P(a, t|x, t_0)+\mu P(b, t|x, t_0)$

and

$\partial_t P(b, t|x, t_0)=\lambda P(a, t|x, t_0)-\mu P(b, t|x, t_0).$

The process is also known under the names Kac process[1] , dichotomous random process.[2]

## Properties

Knowledge of an initial state decays exponentially. Therefore for a time in the remote future, the process will reach the following stationary values, denoted by subscript s:

Mean:

$\langle X \rangle_s = \frac {a\mu+b\lambda}{\mu+\lambda}.$

Variance:

$\operatorname{var} \{ X \}_s = \frac {(a-b)^2\mu\lambda}{(\mu+\lambda)^2}.$

One can also calculate a correlation function:

$\langle X(t),X(s)\rangle_s = \exp(-(\lambda+\mu)|t-s|) \operatorname{var} \{ X \}_s.$

## Application

This random process finds wide application in model building: