Probability · step 8 of 13

Geometric & Poisson distributions

The geometric distribution counts trials until the first success: P(X = k) = (1−p)^(k−1)p, with mean 1/p — so a 1-in-6 event takes 6 attempts on average. The Poisson models counts of rare events in a fixed interval: P(X = k) = e^(−λ)λ^k/k!, with mean AND variance both equal to λ, which is its signature property.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Recognise the geometric setupRepeated independent trials, counting until the first success. The mean is 1/p and the distribution is memoryless.
Recognise the Poisson setupCounts in a fixed time or space, events independent and at a constant average rate λ.
Scale λ with the intervalIf the rate is 3 per hour, then λ = 6 for two hours. Failing to rescale is the classic error.
Use complements for tail probabilitiesP(X ≥ 1) = 1 − e^(−λ) for Poisson; P(X > n) = (1−p)ⁿ for geometric.

Worked example

Calls arrive at a switchboard at 3 per hour, Poisson distributed. Find P(exactly 2 calls in 40 minutes).

40 minutes is 2/3 of an hour, so λ = 3 × (2/3) = 2.
P(X = 2) = e^(−2)·2²/2!.
= e^(−2)·4/2 = 2e^(−2).
Numerically 2 × 0.1353.

Answer. P(X = 2) = 2e^(−2) ≈ 0.271.

Where marks get dropped

These are the specific errors that cost credit on geometric & poisson distributions questions — QED's rubric penalises each of them separately.

Forgetting to rescale λ for the actual interval. The rate and the parameter coincide only for a unit interval.
Using the geometric distribution for a fixed number of trials. That is binomial — geometric has no fixed n.
Confusing the two geometric conventions: counting trials INCLUDING the success (mean 1/p) versus failures BEFORE it (mean (1−p)/p).

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Geometric & Poisson distributions — frequently asked questions

What does memorylessness mean?

P(X > m + n | X > m) = P(X > n): past failures do not improve your future prospects. The geometric is the only discrete distribution with this property.

When does the Poisson approximate the binomial?

When n is large and p small with λ = np moderate. The rule of thumb is n ≥ 50 and np ≤ 10.

Why does the Poisson have variance equal to its mean?

It follows from the derivation as a limit of binomials: np(1−p) → λ as p → 0 with np = λ fixed. A sample variance far from the mean is evidence against a Poisson model.

The rest of Probability

Events, conditional probability, random variables. Each subtopic below has its own method, worked example and mark-losing traps.

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