satunnaiskulun

Satunnaiskulku, often translated as random walk, is a mathematical object that describes a path consisting of a succession of random steps. Imagine a particle starting at a point and then moving to a new position based on a random choice. This process is repeated, creating a path that is unpredictable. The simplest form is the one-dimensional random walk, where a particle moves left or right with equal probability. In two dimensions, the particle can move up, down, left, or right.

The concept of random walks has applications in various fields. In physics, it's used to model phenomena like Brownian motion, the erratic movement of particles suspended in a fluid. In computer science, it's employed in algorithms like PageRank, used by Google to rank web pages. It also appears in finance to model stock prices, and in biology to study the movement of molecules within cells.

Key properties of a random walk include its expected distance from the origin and its variance. For a simple symmetric random walk, the expected distance from the origin after n steps is zero, but the mean squared displacement grows linearly with n. This means that on average, the walk tends to drift away from its starting point over time, but the specific path is highly irregular. The concept can be extended to more complex spaces and with different types of steps.