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Replacing a simulated quantity $H$ by $E[H\mid \mathcal{G}]$ preserves its expectation and reduces variance by the law of total variance. This is Rao-Blackwellization. For a random sum, conditioning on the count and all but one summand replaces an event indicator by a cumulative distribution function evaluation. The resulting estimator needs that conditional expectation to be computable.

Independence factors the events that both variables exceed a threshold or both lie below it. This gives the two cumulative distribution functions above, including for distributions with atoms. The minimum formula is most naturally expressed through survival probabilities, while the maximum formula uses the CDFs directly.

For scale $\sigma >0$ and shape $\xi$, the generalized Pareto distribution on nonnegative excesses has cumulative distribution functionThe case $\xi =0$ is the continuous limit $1-e^{-y/\sigma }$. Positive $\xi$ gives a power-law tail, zero gives an exponential distribution, and negative $\xi$ gives a finite upper endpoint $-\sigma /\xi$. Its role in modelling high-threshold excesses is explained by the Pickands-Balkema-de Haan theorem.

The Kolmogorov distance between two probability distributions with cumulative distribution functions $F,G$ is ${\sup }_{z\in \mathbb{R}}|F(z)-G(z)|$.

If a cumulative distribution function is constant between consecutive lattice points spaced by $h$ near zero, then its Kolmogorov distance from the standard normal distribution function is at least half the increase of that function across one such interval. This gives a lower bound of order $h$.

For positive summable weights $w_j$ at points $q_j$, the strict cumulative function $F(x)={\sum }_{q_j<x}{w}_j$ is nondecreasing and left-continuous. A finite-head/tail argument proves left-continuity, and its right jump at an atom is the mass there. This strict convention differs at atoms from the usual right-continuous cumulative distribution function.

For a finite first absolute moment, the minimizers of expected absolute loss are the medians. With a density, differentiating the loss gives $2F(a)-1$, so every point where the cumulative distribution function equals one half minimizes it. A density gap may give a whole minimizing interval. In contrast, finite squared loss is uniquely minimized by the expectation.

For an atomless measure $\mu$ on $\mathbb{R}$ with cumulative distribution function $F$, the monotone transport to a target with quantile function $G^{-1}$ is $T=G^{-1}\circ F$, defined $\mu$-almost everywhere. It minimizes well-defined costs $d(x-y)$ for convex continuous $d$. With atoms in the source, the common-quantile transport plan $(F^{-1},G^{-1}{)}_{\#}\mathcal{U}(0,1)$ remains available but need not be induced by a map.

A non-atomic measure contains no atom of a measure. For a finite Borel measure on a Polish space, this is equivalent to giving every singleton mass zero. For a probability measure on $\mathbb{R}$, it is also equivalent to continuity of its cumulative distribution function.