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Index/Probability & statistics/PRB-01

PRB-01

Standard normal φ

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ProbabilityGaussian

Governing equation

varphi(z)=e−z2/2/sqrt2pi\\varphi(z)=e^{-z^2/2}/\\sqrt{2\\pi}

z∈[−4,4]z \in \left[-4, 4\right]

where

z
z (—)
\varphi
φ (—)

The figure, part by part

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PRB-01 · curve
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①

Governing law — red

varphi(z)=e−z2/2/sqrt2pi\\varphi(z)=e^{-z^2/2}/\\sqrt{2\\pi}

z∈[−4,4]z \in \left[-4, 4\right]

z=−4z = -4φ=1.34×10−4\varphi = 1.34\times 10^{-4}

②

Domain of the variable — blue

zmin⁡=−4,zmax⁡=4z_{\min} = -4,\quad z_{\max} = 4

z∈[−4,4]z \in \left[-4, 4\right]

z=−4z = -4z=−4z = -4

③

Response curve — green

varphi(z)=e−z2/2/sqrt2pi\\varphi(z)=e^{-z^2/2}/\\sqrt{2\\pi}

z∈[−4,4]z \in \left[-4, 4\right]

z=−4z = -4φ=1.34×10−4\varphi = 1.34\times 10^{-4}

④

Rate of change — amber

dφdz\frac{\mathrm{d}\varphi}{\mathrm{d}z}

z∈[−4,4]z \in \left[-4, 4\right]

z=−3.95z = -3.95φ′=6.49×10−4\varphi^{\prime} = 6.49\times 10^{-4}

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-01 — Standard normal φ) is the form associated with Gaussian. Working symbols: zz →\rightarrow φ\varphi. pdf.

Purpose

Purpose: compute φ\varphi from zz in Probability & statistics via varphi(z)=e−z2/2/sqrt2pi\\varphi(z)=e^{-z^2/2}/\\sqrt{2\\pi} pdf. 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 z=0.000 —z = 0.000\,\mathrm{—}, the governing relation varphi(z)=e−z2/2/sqrt2pi\\varphi(z)=e^{-z^2/2}/\\sqrt{2\\pi} yields φ=0.39894 —\varphi = 0.39894\,\mathrm{—}. pdf. Move a slider: the numbers are this situation, not a canned story.

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Inputs

Outputs

  • φ \varphi0.39894 —
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PRB-01 · curve
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Narration of this film

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