A random walk is a mathematical concept that models the movement of a particle or point in a random or unpredictable manner. It involves taking steps in random directions and can be used to model various phenomena in physics, finance, and other fields.
The expected value of a random walk is the average value that the particle or point is expected to reach after a certain number of steps. It is calculated by multiplying the probability of each step by the value of that step and summing them all together.
The variance of a random walk is a measure of how much the particle or point deviates from its expected value. It is calculated by taking the square of the difference between each step and the expected value, multiplying it by the probability of that step, and summing them all together.
The limit n→∞ represents the idea of taking an infinite number of steps in a random walk. It allows us to analyze the behavior of the random walk over a long period of time and make predictions about its future movements.
The expected value and variance of a random walk become more accurate as the limit n→∞ is approached. This means that the longer the random walk is simulated, the better we can predict the behavior of the particle or point. In some cases, as n→∞, the expected value and variance may converge to a certain value, which can provide insights into the long-term behavior of the random walk.