chebyshev_bound

std.stats.chebyshev_bound · Level L1

Chebyshev's inequality on a finite law, for ε > 0: the actual tail P(|X − μ| ≥ ε), decided as (X − μ)² ≥ ε², and the bound Var(X)/ε². It is Markov's inequality applied to the squared deviation; it needs no independence, and the bound is not the tail.

P(|X − μ| ≥ ε) ≤ Var(X)/ε²

Signature

chebyshev_bound(p: f64[n], x: f64[n], eps: f64[]) → f64[], f64[]

Structure

The function as NOVA stores it: one box per input, operation and output, and arrows that carry values. A double border marks another library function this one runs — called once, or by Scan once per element; select it to open that function.

input
operation
constant
call
output

Verification

Signature proven by NOVA’s shape solver, for every size.
Equal to the reference the mass of the outcomes with (x − μ)² ≥ ε², and the variance over ε² in exact rational arithmetic, on all 40 test cases.
All 80 float64 results inside the running error bound; the closest uses 20% of it.
Interpreter and NumPy backend return bit-identical results.
At float32 too: the 40 test cases rounded to float32, and 40 of them inside the running error bound at u = 2⁻²⁴; the closest uses 17% of it.

Accuracy in detail

correctly rounded (the float64 nearest the exact value): 81%
bit-equal to the NumPy formula in float64: 91%
largest error, in units in the last place: 1.42

Identity

Calls

dot

Called by

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sha256:e35ece37d462711cda3615b37994a6435bccc60cfca20db2753078e052b2a9f5

The semantic hash of the graph. It changes when the program changes, and never when only its documentation does.

Control handle

Symbol: Ω:std.stats.chebyshev_bound · Ω:chebyshev_bound
Pins: sha256:b7691c569211dc52335a30d5dc970c9991e0119c2cdda8df6b7317a634e4009athis graph and the 1 it reaches through calls
Evidence: sha256:a9c3d2c46f653dcaf58f663d929cb48247aed84bc70a28cae23689b5bddc2d27the hash of its verification record
Needs: no capability: a pure function

Through NOVA’s control layer, the symbol launches this function only while the program still matches what it pins: a change to this graph, or to any graph it reaches, needs a migration first.