Random walk

Random Walk A random walk is a stochastic process that models the movement of a point in a 2D plane over time. Imagine a person standing at a random locatio...

Random Walk

A random walk is a stochastic process that models the movement of a point in a 2D plane over time. Imagine a person standing at a random location in a city map. Each step they take will be independent and uniformly distributed, meaning they will be equally likely to move left or right or up or down. Over time, this random movement will lead them to explore the entire city map.

Formal Definition:

A random walk is a sequence of iid (independent and identically distributed) random variables, where the random variables are defined on a 2D plane. Each variable represents the position of the point at a specific time step.

Properties:

A random walk is a Markov process, meaning the current state depends only on the previous state.
It is a stochastic process, meaning the state of the point at a future time step depends not only on the current state but also on the sequence of steps taken in the past.
The expected displacement of a random walk is always zero.
A random walk converges to a unique distribution as time increases.

Example:

Imagine a point walking randomly on a circular board. At any given time, the point is equally likely to be on any other part of the board. This is a simple example of a random walk.

Applications:

Random walks have diverse applications in various fields, including:

Computer science: Random walks are used in algorithms such as Monte Carlo simulations, which are used to solve problems intractable for exact methods.
Statistics: They are used in Bayesian inference and Markov chain analysis.
Physics: Random walks are used to model diffusion processes and chemical reactions.
Economics: They are used to model stock market movements and economic cycles