# RandomWalkProcess

represents a random walk on a line with the probability of a positive unit step p and the probability of a negative unit step 1-p.

RandomWalkProcess[p,q]

represents a random walk with the probability of a positive unit step p, the probability of a negative unit step q, and the probability of a zero step 1-p-q.

# Examples

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## Basic Examples(3)

Simulate a one-dimensional random walk:

For a three-step random walk:

Mean and variance functions:

For a three-step random walk:

Covariance function:

For a three-step random walk:

## Scope(11)

### Basic Uses(5)

Simulate an ensemble of random paths:

Compare paths for different values of the process parameter:

Process parameter estimation:

Estimate the process parameters from sample data:

Correlation function:

For a three-step random walk:

Absolute correlation function:

For a three-step random walk:

### Process Slice Properties(6)

Univariate SliceDistribution:

Univariate probability density:

For a three-step random walk:

Multi-time slice distribution:

Higher-order PDF:

Compute an expectation of an event:

Calculate the probability of an event:

Skewness:

Simple random walk is symmetric for p=1/2:

Limiting values:

For a three-step random walk:

Find values of parameters for which a three-step random walk is symmetric:

Kurtosis:

Find values of the parameter for which a simple random walk is mesokurtic:

Limiting values:

For a three-step random walk:

Find values of the parameters for which a three-step random walk is mesokurtic:

Moment has no closed form for symbolic order:

Generating functions:

Central moment generating function:

FactorialMoment and its generating function:

Cumulant-generating function:

## Applications(1)

A particle starts at the origin and moves to the right by one unit with probability and to the left by one unit with probability after each second. Find the probability that it has moved to the right by four units after 20 seconds:

Movement to the right is denoted by 1 and to the left by -1:

Probability that the particle has moved four units to the right after 20 seconds:

## Properties & Relations(6)

A symmetric 3-step random walk simplifies to a 2-step random walk:

RandomWalkProcess is not weakly stationary:

Transition probability:

The correlation function of a random walk process is the same as of WienerProcess:

Univariate slice distribution is related to BinomialDistribution:

Cumulative distribution function:

Compare with the CDF of the TransformedDistribution of a binomial distribution:

Simulate the proportion of the time spent on the positive side by a symmetric random walk:

Calculate the ratio of time spent on the positive side:

In the limit the ratio has ArcSinDistribution:

## Neat Examples(2)

A symmetric random walk in 2D:

In 3D:

Simulate 500 paths from a random walk process:

Take a slice at 50 and visualize its distribution:

Plot paths and histogram distribution of the slice distribution at 50:

Wolfram Research (2012), RandomWalkProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/RandomWalkProcess.html.

#### Text

Wolfram Research (2012), RandomWalkProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/RandomWalkProcess.html.

#### CMS

Wolfram Language. 2012. "RandomWalkProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RandomWalkProcess.html.

#### APA

Wolfram Language. (2012). RandomWalkProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RandomWalkProcess.html

#### BibTeX

@misc{reference.wolfram_2024_randomwalkprocess, author="Wolfram Research", title="{RandomWalkProcess}", year="2012", howpublished="\url{https://reference.wolfram.com/language/ref/RandomWalkProcess.html}", note=[Accessed: 17-June-2024 ]}

#### BibLaTeX

@online{reference.wolfram_2024_randomwalkprocess, organization={Wolfram Research}, title={RandomWalkProcess}, year={2012}, url={https://reference.wolfram.com/language/ref/RandomWalkProcess.html}, note=[Accessed: 17-June-2024 ]}