Cumulative distribution function ($F_X(x)$)

The cumulative distribution function of a real random variable $X$ isIt is nondecreasing, right-continuous, tends to zero at $-\infty$, and tends to one at $+\infty$.

For $F_{\mu }(t)=\mu ((-\infty ,t])$, every difference quotient at $x$ is nonnegative and bounded by $4\mu (B(x,2|h|))/{\lambda }_1(B(x,2|h|))$. The spherical derivative of a singular measure vanishes, so the two-sided derivative is zero Lebesgue almost everywhere. A singular distribution function need not be constant: differentiation almost everywhere does not recover increments without absolute continuity of a function of the function.

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 $h$.

If $X$ has a continuous cumulative distribution function $F$, then $F(X)$ has the uniform distribution on $(0,1)$. Combined with inverse transform sampling, this constructs the monotone rearrangement $G^{-1}(F(X))$ to a target with quantile function $G^{-1}$.

For correct continuous conditional forecasts, the conditional probability integral transform is uniform given the entire past. Iterating conditional probabilities makes the transformed sequence independent and uniform. Uniformity and serial checks assess absolute predictive calibration using only the forecast distributions and observations.

If distribution functions $F_n$ converge pointwise to a continuous distribution function $F$, then the convergence is uniform. Choose finitely many points whose successive $F$-increments and two tails are small, then use monotonicity to squeeze $F_n-F$ between its values at adjacent points.

For real observations $X_1,\dots ,X_n$, the empirical distribution function is