A drunk man standing on a cliff, takes steps randomly left and right. Each step he takes has a probability of going left and a probability of going right and the size of each step is same. If the drunk man is allowed to randomly step indefinitely, what will be the probability that he falls off the cliff?
Any guesses? Well, let’s again have a glimpse of this problem through “Random Walk”.
The Random Walk theory is based on the irregular motion of the individual pollen particles, studied by botanist, Mr. Robert Brown in 1828. In the process of researching on a random walk, scientists like Einstein and Smoluchowski studied similar subjects like random process, random noise, spectral analysis, and stochastic equations. The first simple model of Random Walk proposed was uncorrelated and unbiased.
- Uncorrelated means the direction of movement is completely independent of the previous directions taken.
- **_Unbiased _**means there is no preferred direction, the direction moved at each step is completely random.
“A random walk is a mathematical object, known as a stochastic or random process, that describes a path that consists of a succession of random steps on some mathematical space such as the integers.” (Source: Wikipedia)
It is a problem, which is closely related to Brownian motion.
Types of Random Walks
1. Correlated Random Walks (CRWs)
It involves a correlation between successive step orientations. This correlation is termed as **Persistence_. _**This produces a local directional bias, each step tends to point in the same direction as the previous one, although the influence of the initial direction of motion progressively diminishes over time and step, orientations are uniformly distributed in the long term. The nature of the motion of animals is similar, hence, CRWs have been constantly used to monitor motion paths of animals in various contexts.
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