6.1 The Poisson distribution

If events occur independently at constant mean rate λ per interval, X~Po(λ) and P(X=r)=e^{−λ}λ^r/r!.

Match λ to the interval length, check that events are countable and rare enough for the model, and use complements for “at least one” questions.

If a call centre averages 3 calls per minute, P(2 calls in one minute)=e^{−3}3²/2.

Changing the interval changes λ proportionally; it is not a universal parameter for every time window.

For X~Po(λ), E(X)=λ and Var(X)=λ. For a time or area interval scaled by k, the mean becomes kλ under a constant-rate model.

Use the equality as a model check, not as a statement that every sample has equal mean and variance. Estimate λ from appropriate exposure.

If the observed average is 4 events per hour, a two-hour interval has Po(8), not Po(4).

Sample variance need not equal sample mean exactly; the equality describes the distributional parameter.

Independent Poisson variables add to a Poisson variable: if X~Po(λ₁) and Y~Po(λ₂), then X+Y~Po(λ₁+λ₂).

Check independence and compatible exposure definitions before combining. For a difference or conditional count, the simple sum rule does not apply.

Counts from two independent sensors with means 2 and 3 per hour combine to Po(5) per hour.

Adding observed counts is always possible, but adding Poisson distributions requires independence and parameter interpretation.

Bin(n,p) has a fixed number n of independent trials with success probability p; Po(λ) models a count in an interval with a rate. They answer different sampling questions.

Use binomial when trials are explicit and bounded; use Poisson when exposure and event rate are natural. A Poisson approximation to binomial needs large n, small p and λ=np.

Ten quality checks with defect probability 0.02 are binomial; defects over a long production interval may be modelled Poisson with the measured rate.

A small p alone does not justify Poisson approximation; n and the product np also matter.

A binomial or Poisson count may be approximated by N(μ,σ²) when its distribution is sufficiently spread. Convert integer events to intervals using continuity correction, such as P(X≤k)≈P(Y<k+0.5).

State μ and σ² from the original model, standardise the corrected boundary and compare the approximation with an exact calculation when accuracy matters.

P(X≤10) becomes P(Y<10.5), not P(Y≤10), under a continuous normal approximation.

The normal variable is continuous, so forgetting the half-unit correction can materially shift a tail probability.