PRB-04

Poisson P(k)

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ProbabilityPoisson

Governing equation

P(k)=e^{-\\lambda}\\lambda^k/k!

$\lambda \in \left[0.1, 15\right]$

where

\lambda: λ (—)
k: k (—)
P: P (—)

The figure, part by part

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①

Governing law — red

P(k)=e^{-\\lambda}\\lambda^k/k!

$\lambda \in \left[0.1, 15\right]$

$\lambda = 0.1$ $P = 1.51\times 10^{-4}$

Along the curve

②

Domain of the variable — blue

\lambda_{\min} = 0.1,\quad \lambda_{\max} = 15

$\lambda \in \left[0.1, 15\right]$

$\lambda = 0.1$ $\lambda = 0.1$

Along the curve

③

Response curve — green

P(k)=e^{-\\lambda}\\lambda^k/k!

$\lambda \in \left[0.1, 15\right]$

$\lambda = 0.1$ $P = 1.51\times 10^{-4}$

Along the curve

④

Rate of change — amber

\frac{\mathrm{d}P}{\mathrm{d}\lambda}

$\lambda \in \left[0.1, 15\right]$

$\lambda = 0.194$ $P^{\prime} = 0.015$

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-04 — Poisson P(k)) is the form associated with Poisson. Working symbols: $\lambda$ , $k$ $\rightarrow$ $P$ . Counts.

Purpose

Purpose: compute

P

from

\lambda

k

in Probability & statistics via

P(k)=e^{-\\lambda}\\lambda^k/k!

Counts. 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

\lambda = 3.000\,\mathrm{—}

k = 3.000\,\mathrm{—}

, the governing relation

P(k)=e^{-\\lambda}\\lambda^k/k!

yields

P = 0.22404\,\mathrm{—}

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Calculator

Inputs

λ \lambda3 —k k3 —

Outputs

P P0.22404 —

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