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:

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:

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:

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: