PRB-21

Exponential mean

Mean waiting time.

Reading speed

GoverningExponential mean

Governing equation

\\mu=1/\\lambda

$lam \in \left[0.02, 4\right]$

where

lam: lam (1/s)
mu: Exponential mean (s)

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=1/\\lambda

$lam \in \left[0.02, 4\right]$

$lam = 0.02$ $mu = 50$

Along the curve

②

Domain of the variable — blue

lam_{\min} = 0.02,\quad lam_{\max} = 4

$lam \in \left[0.02, 4\right]$

$lam = 0.02$ $lam = 0.02$

Along the curve

③

Response curve — green

\\mu=1/\\lambda

$lam \in \left[0.02, 4\right]$

$lam = 0.02$ $mu = 50$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}mu}{\mathrm{d}lam}

$lam \in \left[0.02, 4\right]$

$lam = 0.045$ $mu^{\prime} = -713.6$

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-21 — Exponential mean) is the form associated with Exponential mean. Working symbols: $lam$ $\rightarrow$ $mu$ . Mean waiting time. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

mu

from

lam

in Probability & statistics via

\\mu=1/\\lambda

Mean waiting time. 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

lam = 0.200\,\mathrm{1/s}

, the governing relation

\\mu=1/\\lambda

yields

mu = 5.000\,\mathrm{s}

. 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

lam lam0.2 1/s

Outputs

Exponential mean mu5.000 s

Reading speed

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Narration of this film

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

Mean waiting time. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube