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  "body": "\n## Summary\n\nA Brownian bridge is a Brownian motion on $[0, 1]$ forced to return to $0$ at time $1$.\nIt is a continuous centred [[Gaussian process|22WEM6RTW0]] with covariance $\\min(s, t) - st$.\nIt is the limit of the uniform [[empirical process|XKFG1676CG]], which makes it central in goodness-of-fit tests such as the Kolmogorov–Smirnov test.\n\n## Explanations\n\n### L1 human\n\nImagine a Brownian path that starts at $0$, wanders at random, and is tied down so that it must be back at $0$ at time $1$.\nThis pinned random path is a Brownian bridge.\nA simple recipe: take a Brownian path $W$ and subtract the straight line from $(0, 0)$ to $(1, W_1)$.\nNear the two ends it is almost fixed, and in the middle, at $t = 1/2$, it varies most, with variance $1/4$.\n",
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