PRB-24

Chebyshev bound

Distribution-free tail.

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GoverningChebyshev bound

Governing equation

P(|X-\\mu|\\ge k\\sigma)\\le 1/k^2

$k \in \left[1.1, 6\right]$

where

k: k (—)
b: Chebyshev bound (—)

The figure, part by part

Each numbered marker wears the color of its card. Hover a card to light the marker. Tap a marker to jump to that card. The slider walks a red point along the curve; Animate drawing traces it like a pen.

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①

Governing law — red

P(|X-\\mu|\\ge k\\sigma)\\le 1/k^2

$k \in \left[1.1, 6\right]$

$k = 1.1$ $b = 0.826$

Along the curve

②

Domain of the variable — blue

k_{\min} = 1.1,\quad k_{\max} = 6

$k \in \left[1.1, 6\right]$

$k = 1.1$ $k = 1.1$

Along the curve

③

Response curve — green

P(|X-\\mu|\\ge k\\sigma)\\le 1/k^2

$k \in \left[1.1, 6\right]$

$k = 1.1$ $b = 0.826$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}b}{\mathrm{d}k}

$k \in \left[1.1, 6\right]$

$k = 1.131$ $b^{\prime} = -1.385$

Along the curve

Lecture brief

Historical brief

Bernoulli trials, Poisson, Gauss, Bayes and Shannon entropy turned chance into a calculus of belief and noise. The lab computes likelihood, interval and information in SI-free counts. This sheet (PRB-24 — Chebyshev bound) is the form associated with Chebyshev bound. Working symbols: $k$ $\rightarrow$ $b$ . Distribution-free tail. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

b

from

k

in Probability & statistics via

P(|X-\\mu|\\ge k\\sigma)\\le 1/k^2

Distribution-free tail. Use it when a real probability & statistics question must be answered in SI before a code check.

Live realistic example

In symbols

Live case. Given

k = 2.000\,\mathrm{—}

, the governing relation

P(|X-\\mu|\\ge k\\sigma)\\le 1/k^2

yields

b = 0.250\,\mathrm{—}

. One governing identity, SI units, a single sweep on the sheet. Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

k k2 —

Outputs

Chebyshev bound b0.250 —

Reading speed

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Narration of this film

One governing identity, SI units, a single sweep on the sheet.

Distribution-free tail. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube