# TelegraphProcess

represents a telegraph process with rate μ.

# Details # Examples

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## Basic Examples(3)

Simulate the telegraph process:

Mean and variance functions:

Covariance function:

## Scope(12)

### Basic Uses(6)

Simulate an ensemble of paths:

Simulate with arbitrary precision:

Compare paths for different values of the process parameter:

Process parameter estimation from sample data:

Correlation function:

Absolute correlation function:

### Process Slice Properties(6)

Telegraph process assumes only two values:

Compare the first-order PDFs in time for the two values:

The limiting value is the same for both values:

Second-order PDF:

Check if the PDF sums to 1:

Compute an expectation of an expression:

Calculate the probability of an event:

Skewness is negative:

The limiting values:

Kurtosis is positive:

The limiting values:

Moment-generating functions:

CentralMoment and its generating function:

FactorialMoment and its generating function:

Cumulant-generating function:

## Applications(1)

The collision times for a particle moving between two barriers are distributed exponentially, with a mean of 3 microseconds. If the particle starts at the right-hand barrier ( ) and moves towards the left-hand barrier ( ), then simulate the collision process for 100 microseconds:

Variance for the slice distribution:

Compare with the variance of a random sample:

## Properties & Relations(6)

TelegraphProcess is a jump process:

The telegraph process is not weakly stationary:

The absolute correlation depends only on time differences:

But the mean function is not constant:

It is, however, asymptotically weakly stationary:

The number of jumps at time t follows a PoissonDistribution:

Generate a random sample of paths from a telegraph process and record their lengths:

The times between jumps follow an ExponentialDistribution:

Generate a random sample of paths from a telegraph process:

Transition probability:

TelegraphProcess is a transformation of a PoissonProcess:

Simulate the process:

Probability density function for a time slice of the process:

Compare with the PDF for TelegraphProcess:

Compare CovarianceFunction:

## Neat Examples(1)

Simulate paths from a telegraph process:

Take a slice at 20 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 20:

Introduced in 2012
(9.0)