In probability theory

Probability theory

Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...

, the telegraph process is a memoryless

Memorylessness

In probability and statistics, memorylessness is a property of certain probability distributions: the exponential distributions of non-negative real numbers and the geometric distributions of non-negative integers....

 continuous-time stochastic process

Stochastic process

In probability theory, a stochastic process , or sometimes random process, is the counterpart to a deterministic process...

 that shows two distinct values.

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

Master equation

In physics and chemistry and related fields, master equations are used to describe the time-evolution of a system that can be modelled as being in exactly one of countable number of states at any given time, and where switching between states is treated probabilistically...

s:


and


The process is also known under the names Kac

Mark Kac

Mark Kac was a Polish mathematician. His main interest was probability theory. His question, "Can one hear the shape of a drum?" set off research into spectral theory, with the idea of understanding the extent to which the spectrum allows one to read back the geometry. Kac completed his Ph.D...

 process
, dichotomous random process.

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:

Variance:

One can also calculate a correlation function

Correlation function

A correlation function is the correlation between random variables at two different points in space or time, usually as a function of the spatial or temporal distance between the points...

:

Application

This random process finds wide application in model building

Model building

Model building as a hobby involves the creation of models either from kits or from materials and components acquired by the builder.Categories of modelling include:*Scale model building*Live steam models*Matchstick models*Military models...

:

• In physics
  Physics
  Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
  
  , spin systems
  Spin (physics)
  In quantum mechanics and particle physics, spin is a fundamental characteristic property of elementary particles, composite particles , and atomic nuclei.It is worth noting that the intrinsic property of subatomic particles called spin and discussed in this article, is related in some small ways,...
  
   and fluorescence
  Fluorescence
  Fluorescence is the emission of light by a substance that has absorbed light or other electromagnetic radiation of a different wavelength. It is a form of luminescence. In most cases, emitted light has a longer wavelength, and therefore lower energy, than the absorbed radiation...
  
   intermittency
  Fluorescence intermittency
  Fluorescence intermittency, or blinking, is the phenomenon of random switching between ON and OFF states of the emitter under its continuous excitation...
  
   show dichotomous properties. But especially in single molecule experiments probability distribution
  Probability distribution
  In probability theory, a probability mass, probability density, or probability distribution is a function that describes the probability of a random variable taking certain values....
  
  s featuring algebraic tails are used instead of the exponential distribution
  Exponential distribution
  In probability theory and statistics, the exponential distribution is a family of continuous probability distributions. It describes the time between events in a Poisson process, i.e...
  
   implied in all formulas above.
• In finance
  Finance
  "Finance" is often defined simply as the management of money or “funds” management Modern finance, however, is a family of business activity that includes the origination, marketing, and management of cash and money surrogates through a variety of capital accounts, instruments, and markets created...
  
   for describing stock
  Stock
  The capital stock of a business entity represents the original capital paid into or invested in the business by its founders. It serves as a security for the creditors of a business since it cannot be withdrawn to the detriment of the creditors...
  
  prices