Counting Clicks on a Geiger Counter

Radioactive atoms decay at random, each independently of the others. Count the clicks in one second and you get a Poisson-distributed number: the average rate λ alone gives the chance of every count, and the spread grows like √λ.

How it works

The Poisson distribution

In one sentence

A random variable that counts events occurring independently at a constant average rate $\lambda$ per interval has the Poisson distribution $P(X=k) = \lambda^k e^{-\lambda}/k!$ for $k = 0, 1, 2, \dots$ , with mean and variance both equal to $\lambda$ . It is the limit of the binomial distribution as $n \to \infty$ with $np = \lambda$ held fixed.

General formula

P(X = {\htmlData{sym=1}{k}}) = \frac{{\htmlData{sym=0}{\lambda}}^{\htmlData{sym=1}{k}} e^{-{\htmlData{sym=0}{\lambda}}}}{{\htmlData{sym=1}{k}}!}, \qquad {\htmlData{sym=2}{\mu}} = {\htmlData{sym=0}{\lambda}}, \quad {\htmlData{sym=3}{\sigma}} = \sqrt{\htmlData{sym=0}{{\lambda}}}

Many independent chances for a rare event add up to an average of λ events, and the exact count still wanders by about √λ.

What the symbols mean

$\lambda$
Average rate
The mean number of clicks per second.
$k$
A count
A possible number of clicks in one second.
$\mu$
Mean
The long-run average count, equal to λ.
$\sigma$
Standard deviation
The typical swing of the count around λ: the square root of λ.

Split a second into many tiny moments. In each, a click is rare and independent of the others, so the count is binomial with a huge $n$ and a tiny $p$ ; in the limit it becomes Poisson with $\lambda = np$ .
$\lambda$ is both the mean count and its variance, so the standard deviation is $\sqrt{\lambda}$ .
For small $\lambda$ the distribution is skewed right (zero clicks is common); for large $\lambda$ it looks normal, with mean $\lambda$ and standard deviation $\sqrt{\lambda}$ .
Because the spread grows only like $\sqrt{\lambda}$ , counting longer measures a rate more precisely: the relative error is $1/\sqrt{\lambda}$ .

Learn the math

The Poisson DistributionStats · Lesson 17, Discrete Random Variables Counting Defects: Binomial and PoissonEng Stats · Lesson 4, Probability Models for Engineering Time to Failure: The Exponential ModelEng Stats · Lesson 5, Probability Models for Engineering p and c Charts for DefectsEng Stats · Lesson 28, Statistical Process Control

Learn the math

The Poisson DistributionStats · Lesson 17, Discrete Random Variables

Counting Defects: Binomial and PoissonEng Stats · Lesson 4, Probability Models for Engineering

Time to Failure: The Exponential ModelEng Stats · Lesson 5, Probability Models for Engineering

p and c Charts for DefectsEng Stats · Lesson 28, Statistical Process Control