PRB-22

Geometric mean

Trials until first success.

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GoverningGeometric mean

Governing equation

E[X]=1/p

$p \in \left[0.02, 0.9\right]$

where

p: p (—)
EX: Geometric mean (—)

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

E[X]=1/p

$p \in \left[0.02, 0.9\right]$

$p = 0.02$ $EX = 50$

Along the curve

②

Domain of the variable — blue

p_{\min} = 0.02,\quad p_{\max} = 0.9

$p \in \left[0.02, 0.9\right]$

$p = 0.02$ $p = 0.02$

Along the curve

③

Response curve — green

E[X]=1/p

$p \in \left[0.02, 0.9\right]$

$p = 0.02$ $EX = 50$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}EX}{\mathrm{d}p}

$p \in \left[0.02, 0.9\right]$

$p = 0.026$ $EX^{\prime} = -1609.3$

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-22 — Geometric mean) is the form associated with Geometric mean. Working symbols: $p$ $\rightarrow$ $EX$ . Trials until first success. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

EX

from

p

in Probability & statistics via

E[X]=1/p

Trials until first success. 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

p = 0.200\,\mathrm{—}

, the governing relation

E[X]=1/p

yields

EX = 5.000\,\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

p p0.2 —

Outputs

Geometric mean EX5.000 —

Reading speed

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Narration of this film

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

Trials until first success. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube