HistogramDistribution

The empirical density of $data$ , binned into equal-width bins of width $dx$ — piecewise-constant.

HistogramDistribution(data, {dx})bins of width dx, starting at floor(min(data)/dx)*dx.

HistogramDistribution(data, {x0, x1, dx})bins of width dx over the explicit range [x0, x1].

Typed signatures

Type	Package	Applies
`(list<real>, list<real>) -> distribution`	`statistics`	always

Domain: Statistics

Details

EXPLICIT bins only — Wolfram's own automatic-binning rule ({} with no spec) is not pinned down here and is out of scope. The {dx} form picks $x_0 = \lfloor \min(data)/dx \rfloor \cdot dx$ as the first bin's left edge (a documented choice of this package, not Wolfram's own automatic rule) and covers every data point up to $\max(data)$ ; {x0, x1, dx} gives the range explicitly.
$PDF(x) = count_i / (n \cdot dx)$ for the bin $i$ containing $x$ , $0$ outside every bin — piecewise CONSTANT.
$CDF$ is piecewise LINEAR (the PDF's antiderivative) — full bins below $x$ contribute their whole count, $x$ 's own bin a fractional share.
$Mean = \sum_i count_i \cdot mid_i / n$ ; $Variance = E[X^2] - Mean^2$ , with each bin's $E[X^2\mid bin]$ the standard uniform-on- $[a,b]$ second moment $(a^2+ab+b^2)/3$ .
Bin counts are exact integers and every PDF/CDF/Mean/Variance built from them stays exact (rational arithmetic throughout), even though the BINNING itself (which bin a data point falls in) is computed in plain numbers — inherently a floor/division on a numeric position.