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For $r,s\in [0,1]$ and a random variable $U$ with the uniform distribution on $[0,1]$, set $B_r=1_{\{U\le r\}}$ and $B_s=1_{\{U\le s\}}$. Each has the corresponding Bernoulli distribution, and $\mathbb{P}(B_r\ne B_s)=|r-s|$, because disagreement occurs precisely between the two thresholds. This is a maximal coupling of the two laws.

Independent uniform distribution draws $U_1,U_2$ on $(0,1)$ produce independent standard normal distribution draws $X_1=\sqrt{-2\log U_1}\cos (2\pi U_2)$ and $X_2=\sqrt{-2\log U_1}\sin (2\pi U_2)$. The proof transforms an independent uniform angle and radial density $r{e}^{-r^2/2}$ through the polar-coordinate Jacobian.

A copula is a joint distribution function on a unit cube whose one-dimensional marginal distributions are uniform distributions. Applying each continuous marginal distribution to its random variable produces uniform marginals by the probability integral transform; their copula records the remaining dependence. Independent uniform coordinates and one common uniform coordinate give different copulas with the same marginals.

The hit-and-run sampler moves within a full-dimensional compact convex set by choosing a uniform direction, then a uniform point on the chord through the current point in that direction. Its invariant law is the uniform distribution on the set. Starting in the interior keeps the chain in the interior almost surely. To define absolutely continuous transitions also at boundary points, redraw directions whose chords have zero length. Without this convention, a corner can have a positive holding atom.

Independent centered Laplace distributions for pixel intensities give the prior distributionThis prior favors small intensities through its sharp peak at zero, while its tails permit isolated appreciable pixels. Its negative log density is an L1 norm penalty, so with a Gaussian likelihood a maximum a posteriori estimate minimizes a sum of squared residuals plus that penalty. A draw from this continuous prior has no exactly zero coordinate with positive probability; sparsity of a maximizing estimate and concentration of prior mass near zero are different statements. The product prior also does not encode spatial contiguity. Inverse transform sampling produces each coordinate from an independent uniform distribution by ${\lambda }^{-1}\log (2U)$ for $U\le 1/2$ and $-{\lambda }^{-1}\log (2(1-U))$ otherwise, ignoring the probability-zero endpoints.

Among probability distributions on a finite set of $m$ elements, the uniform distribution uniquely maximizes information entropy, attaining $\log m$.