PRB-26

Beta mean

Beta distribution mean.

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GoverningBeta mean

Governing equation

\\mu=\\alpha/(\\alpha+\\beta)

$al \in \left[0.2, 12\right]$

where

al: al (—)
be: be (—)
mu: Beta 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

\\mu=\\alpha/(\\alpha+\\beta)

$al \in \left[0.2, 12\right]$

$al = 0.2$ $mu = 0.038$

Along the curve

②

Domain of the variable — blue

al_{\min} = 0.2,\quad al_{\max} = 12

$al \in \left[0.2, 12\right]$

$al = 0.2$ $al = 0.2$

Along the curve

③

Response curve — green

\\mu=\\alpha/(\\alpha+\\beta)

$al \in \left[0.2, 12\right]$

$al = 0.2$ $mu = 0.038$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}mu}{\mathrm{d}al}

$al \in \left[0.2, 12\right]$

$al = 0.274$ $mu^{\prime} = 0.18$

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-26 — Beta mean) is the form associated with Beta mean. Working symbols: $al$ , $be$ $\rightarrow$ $mu$ . Beta distribution mean. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

mu

from

al

be

in Probability & statistics via

\\mu=\\alpha/(\\alpha+\\beta)

Beta distribution mean. 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

al = 2.000\,\mathrm{—}

be = 5.000\,\mathrm{—}

, the governing relation

\\mu=\\alpha/(\\alpha+\\beta)

yields

mu = 0.286\,\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

al al2 —be be5 —

Outputs

Beta mean mu0.286 —

Reading speed

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Narration of this film

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

Beta distribution mean. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube