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.
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.
- RandomWalkProcess is also known as a lattice random walk.
- RandomWalkProcess is a discrete-time and discrete-state random process.
- RandomWalkProcess[p] value at time t follows TransformedDistribution[2 x-t,xBinomialDistribution[t,p]].
- RandomWalkProcess allows p and q to be real numbers between 0 and 1 such that p+q≤1.
- RandomWalkProcess can be used with such functions as Mean, PDF, Probability, and RandomFunction.
Examplesopen allclose all
Basic Examples (3)
Basic Uses (5)
Process Slice Properties (6)
Univariate probability density:
Multi-time slice distribution:
Compute an expectation of an event:
Calculate the probability of an event:
Simple random walk is symmetric for p=1/2:
Find values of parameters for which a three-step random walk is symmetric:
Find values of the parameter for which a simple random walk is mesokurtic:
Find values of the parameters for which a three-step random walk is mesokurtic:
Moment has no closed form for symbolic order:
Central moment generating function:
FactorialMoment and its generating function:
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:
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:
Wolfram Research (2012), RandomWalkProcess, Wolfram Language function, https://reference.wolfram.com/language/ref/RandomWalkProcess.html.
Wolfram Language. 2012. "RandomWalkProcess." Wolfram Language & System Documentation Center. Wolfram Research. https://reference.wolfram.com/language/ref/RandomWalkProcess.html.
Wolfram Language. (2012). RandomWalkProcess. Wolfram Language & System Documentation Center. Retrieved from https://reference.wolfram.com/language/ref/RandomWalkProcess.html