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.’ Random walks are an important area of study in many fields, such as biology and finance, and have significant real-world applications. Of particular interest to myself is its importance in probability theory, specifically in regards to the simulation method Markov Chain Monte Carlo (‘MCMC’). In some of my other projects I have utilized the MCMC method to randomly simulate from a sample within a given probability distribution.

This first image depicts the paths of five thousand random walks.

Here we see a histogram of the ending positions of the random walks. The distribution of the random walks is normal.

In this instance I set a window through which only random walks within a certain range at the window’s point could pass through.

And now you see the new distribution of the final position of the random walks.