PRB-03

Binomial mean

Also σ=√(npq).

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Probabilitybinomial mean

Governing equation

\\mu=np

$p \in \left[0.01, 0.99\right]$

where

n: n (—)
p: p (—)
\mu: μ (—)
\sigma: σ (—)

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

$p \in \left[0.01, 0.99\right]$

$p = 0.01$ $\mu = 0.2$

Along the curve

②

Domain of the variable — blue

p_{\min} = 0.01,\quad p_{\max} = 0.99

$p \in \left[0.01, 0.99\right]$

$p = 0.01$ $p = 0.01$

Along the curve

③

Response curve — green

\\mu=np

$p \in \left[0.01, 0.99\right]$

$p = 0.01$ $\mu = 0.2$

Along the curve

④

Rate of change — amber

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

$p \in \left[0.01, 0.99\right]$

$p = 0.016$ $\mu^{\prime} = 20$

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-03 — Binomial mean) is the form associated with binomial mean. Working symbols: $n$ , $p$ $\rightarrow$ $\mu$ , $\sigma$ . Also σ=√(npq).

Purpose

Purpose: compute

\mu

\sigma

from

n

p

in Probability & statistics via

\\mu=np

Also σ=√(npq). 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{—}

\sigma = 2.049\,\mathrm{—}

. Also σ=√(npq). Move a slider: the numbers are this situation, not a canned story.

Calculator

Inputs

n n20 —p p0.3 —

Outputs

μ \mu6.000 —
σ \sigma2.049 —

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Narration of this film

Also σ=√(npq).

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