PRB-27

Gamma mean

Gamma (shape-scale) mean.

Reading speed

GoverningGamma mean

Governing equation

\\mu=k\\theta

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

where

k: k (—)
th: th (—)
mu: Gamma 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=k\\theta

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

$k = 0.2$ $mu = 0.4$

Along the curve

②

Domain of the variable — blue

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

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

$k = 0.2$ $k = 0.2$

Along the curve

③

Response curve — green

\\mu=k\\theta

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

$k = 0.2$ $mu = 0.4$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}mu}{\mathrm{d}k}

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

$k = 0.274$ $mu^{\prime} = 2$

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-27 — Gamma mean) is the form associated with Gamma mean. Working symbols: $k$ , $th$ $\rightarrow$ $mu$ . Gamma (shape-scale) mean. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

mu

from

k

th

in Probability & statistics via

\\mu=k\\theta

Gamma (shape-scale) 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

k = 3.000\,\mathrm{—}

th = 2.000\,\mathrm{—}

, the governing relation

\\mu=k\\theta

yields

mu = 6.000\,\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

k k3 —th th2 —

Outputs

Gamma mean mu6.000 —

Reading speed

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Narration of this film

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

Gamma (shape-scale) mean. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube