Conditioned Brownian motion that returns to zero — the stochastic backbone of statistics
A Brownian bridge B(t) is standard Brownian motion W(t) conditioned on W(1)=0. Formula: B(t) = W(t) − t·W(1). Variance: Var(B(t)) = t(1−t), which is zero at both endpoints. Appears in the Kolmogorov-Smirnov test: the empirical CDF deviation converges to |B(t)| = Brownian bridge in distribution. The KS mode shows the KS statistic D_n = sup|F_n(x)−F(x)|. Arcsine law: time spent positive follows arcsin distribution.