Random walks · Brownian motion · Geometric BM · Poisson process · CLT
Explore random walks, geometric Brownian motion (stock prices), and Poisson processes. See how the Central Limit Theorem makes the sum of independent random steps converge to a Gaussian distribution regardless of the step distribution.
A 1D random walk X_n = Σε_i where each ε_i = ±1 with equal probability. After n steps the standard deviation grows as σ = √n. The distribution of endpoints converges to N(0,n) by the Central Limit Theorem. In 2D, the walker covers area proportional to n but returns to origin infinitely often (recurrence).
Stock prices follow S(t) = S₀·exp((μ−σ²/2)t + σW_t) where W_t is a Wiener process. The log-normal distribution means prices can't go negative. The Itô term −σ²/2 corrects for the non-linearity of exponentiation. 95% confidence bands grow as ±2σ√t over time.
Events arrive at constant rate λ. The number of events in time t follows Poisson(λt): P(N=k) = (λt)^k e^(−λt)/k!. Inter-arrival times are exponential with mean 1/λ. Applications: radioactive decay, network packets, insurance claims. The Poisson process is the unique process with stationary independent increments.
Switch between 1D Walk, 2D Walk, Stock Price (geometric BM), and Poisson tabs. Adjust N paths, steps, drift μ, volatility σ, and arrival rate λ. Click Regenerate to resample all paths. The histogram shows the distribution of final positions. Watch statistics update — notice variance grows linearly with steps in a random walk.