Random Walks

Mathematical Modeling · Axiom Academy

Unit 4: Probabilistic Models - Mathematical Modeling The simplest random walk takes place on the integers. Starting at position S_0 = 0, at each time step n, the walker takes a step X_n that is +1 (right) with probability p or -1 (left) with probability q = 1 - p. 2. Properties: Mean and Variance For a random walk, we can compute the expected position and how spread out the walker's position becomes over time. The mean tells us the average position, while the variance measures the typical spread. Expected value of each step: E[X_i] = p - q Variance of each step: Var(X_i) = 4pq For symmetric walk (p = 1/2): E[S_n] = 0, Var(S_n) = n One of the most famous applications of random walks is the gambler's ruin problem. A gambler starts with k dollars and plays a series of bets, winning or losing 1 each round. The game ends when reaching N dollars (wins) or 0 dollars (ruined). 4. Applications of Random Walks Random walks appear throughout science and engineering, providing models for systems where randomness plays a fundamental role: Stock Prices: The random walk hypothesis suggests price changes are unpredictable Molecular Diffusion: Particles in fluid undergo Brownian motion Polymer Chains: Each monomer link is a random step Search Algorithms: PageRank uses random walks on web graphs

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