PRB-33

Binomial mean

np successes in expectation.

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GoverningBinomial mean

Governing equation

\\mu=np

$p \in \left[0.02, 0.98\right]$

where

n: n (—)
p: p (—)
mu: Binomial 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=np

$p \in \left[0.02, 0.98\right]$

$p = 0.02$ $mu = 0.4$

Along the curve

②

Domain of the variable — blue

p_{\min} = 0.02,\quad p_{\max} = 0.98

$p \in \left[0.02, 0.98\right]$

$p = 0.02$ $p = 0.02$

Along the curve

③

Response curve — green

\\mu=np

$p \in \left[0.02, 0.98\right]$

$p = 0.02$ $mu = 0.4$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}mu}{\mathrm{d}p}

$p \in \left[0.02, 0.98\right]$

$p = 0.026$ $mu^{\prime} = 20$

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-33 — Binomial mean) is the form associated with Binomial mean. Working symbols: $n$ , $p$ $\rightarrow$ $mu$ . np successes in expectation. Pedagogical SI sheet with a live model and a swept parameter.

Purpose

Purpose: compute

mu

from

n

p

in Probability & statistics via

\\mu=np

np successes in expectation. 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

n = 20.000\,\mathrm{—}

p = 0.300\,\mathrm{—}

, the governing relation

\\mu=np

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

n n20 —p p0.3 —

Outputs

Binomial mean mu6.000 —

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Narration of this film

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

np successes in expectation. Pedagogical SI sheet with a live model and a swept parameter.

Reading speed

Watch on YouTube