Other Probability Distributions: Binomial & Poisson

Info sheet · Statistics for Psychology & Neuroscience

Author

Andrew Bell

Published

August 13, 2026

Info sheet 0.2 (draft) · Prerequisites: probability basics; the idea of a distribution · Give feedback ↗

Working notes for the author — not shown to students once collapsed; remove before publishing.

  • [need to split poisson and binomial into separate sheets]
  • use coin flip, dice role interactive element for binomial
  • introduce with replacement/without replacement, etc.

What you’ll get from this sheet

Not everything is a bell curve. Two workhorse distributions for count data. By the end you should be able to:

  1. Use the binomial for a fixed number of yes/no trials, and the Poisson for counts of events.
  2. State each one’s mean and variance, and how they connect.

The binomial counts successes in a fixed number \(n\) of independent yes/no trials with probability \(p\): mean \(np\), variance \(np(1-p)\). The Poisson counts events in a fixed window at average rate \(\lambda\): mean = variance = \(\lambda\). For rare events (large \(n\), small \(p\)), the binomial ≈ Poisson with \(\lambda = np\).

Two distributions for counts

Many measurements are counts, not continuous quantities, and the normal curve isn’t the right tool.

  • Binomial — you have a fixed number of trials \(n\), each a success/failure with probability \(p\), and you count the successes: how many of 20 patients respond to a drug, how many of 10 coin flips are heads. Its probability mass function is \(P(X=k)=\binom{n}{k}p^k(1-p)^{n-k}\), with mean \(np\) and variance \(np(1-p)\).
  • Poisson — you count events in a window (time or space) that occur at an average rate \(\lambda\), with no fixed upper limit: spikes from a neuron per second, emails per hour, typos per page. Its PMF is \(P(X=k)=\dfrac{e^{-\lambda}\lambda^k}{k!}\), and its defining quirk is that its mean and variance are both \(\lambda\).

Explore the shapes

Pick a distribution and move its parameters — watch the bars, the mean, and the variance respond:

For the binomial, at \(p = 0.5\) the bars are symmetric; push \(p\) toward 0 or 1 and they skew, and the variance \(np(1-p)\) is always less than the mean. For the Poisson, small \(\lambda\) gives a sharply right‑skewed pile near zero that becomes more symmetric as \(\lambda\) grows — and the mean always equals the variance. Set the binomial to large \(n\) and small \(p\) (say \(n=40\), \(p=0.04\)) and its shape becomes almost indistinguishable from a Poisson with \(\lambda = np\): the Poisson is the limit of the binomial for rare events.

NoteIn the wild: counting rare events — Australia’s gun laws

Why does the choice of distribution matter beyond a textbook? Consider mass shootings. They are, mercifully, rare — discrete events happening at some low average rate over years, with no fixed number of “trials.” That’s exactly the setting the Poisson distribution was built for, and exactly where a normal distribution misleads: a bell curve would assign probability to negative and fractional shootings, and would badly misjudge the chance of a long run of zero‑event years.

This is the reasoning behind Simon Chapman and colleagues’ influential study of Australia’s 1996 National Firearms Agreement — the sweeping reforms enacted after the Port Arthur massacre, in which 35 people were killed (Chapman, Alpers & Jones, 2016, JAMA 316(3):291–299). In the 18 years before the reforms Australia recorded 13 fatal mass shootings; in the 20 years after, none. To ask whether that drop is more than chance you don’t compare means with a t-test — you model the counts as a Poisson process (events in non‑overlapping intervals treated as independent Poisson/negative‑binomial variables) and ask how surprising a run of zeros would be under the pre‑reform rate. Match the distribution to the phenomenon — rare‑event counts are Poisson, not normal — and the statistics finally fit the question.

Poisson vs Gaussian: watch the bell curve fail

Same hypothetical count data — an average of λ rare events per period — modelled two ways: the Poisson (correct for counts, green bars) and a normal curve \(N(\lambda,\lambda)\) laid over it (the tempting but wrong “bell curve”, blue line). Drag λ down toward the rare‑event range and watch the Gaussian break.

At a genuinely rare rate — under one event per period, roughly Australia’s pre‑reform mass‑shooting rate of about 0.7 per year — the two models disagree sharply. The Poisson says a zero‑event period is common (\(e^{-\lambda}\)); the normal, forced to be symmetric around λ, understates that zero probability and hangs a slab of probability out over negative counts, which can’t occur. Only when λ is large (common events) do the bars and the bell line up. That mismatch is the whole case for choosing a distribution that matches the data‑generating process — and why rare‑event analyses like the Australian gun‑law studies use Poisson models rather than a normal one.

See it in code

binopdf(8, 20, 0.3)     % binomial P(X = 8)
poisspdf(3, 4)          % Poisson P(X = 3)
[mean(poissrnd(4,1e5,1)), var(poissrnd(4,1e5,1))]  % ~ equal (= lambda)

The R and Python tabs run live; MATLAB is a static reference.

A neuron fires on average 4 times per second. Which distribution models the spike count in one second, and what are its mean and variance?

The Poisson, with \(\lambda = 4\) — it counts events in a fixed window with no upper limit and a constant average rate. Its mean and variance are both 4, so the SD is \(\sqrt{4}=2\). (If instead you had a fixed number of trials with a success probability each — say, 20 stimuli and the chance of a response — you’d use the binomial.) A useful diagnostic: if your count data have variance much bigger than their mean (“overdispersion”), a plain Poisson is too simple.

Match the distribution to the process: binomial needs a fixed \(n\) of yes/no trials; Poisson is for open‑ended counts at a rate. Don’t default to the normal for counts — it allows negative and fractional values and, for small means, fits badly (though for large \(np\) or large \(\lambda\) both approach normal). And real count data are often overdispersed (variance > mean), breaking the Poisson’s mean = variance assumption — reach for a negative‑binomial or a quasi‑Poisson model then.

Where this shows up next

These are the outcome distributions behind logistic (binomial) and Poisson regression — the “generalised” in generalised linear models. See Chapter (Distributions) and the Logistic Regression sheet.