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Statistics & Data Science · pattern A, Numeric fields

Poisson probabilities for counts in an interval.

Enter the mean rate and the count of interest for the exact and cumulative probabilities, with the mean and variance that make the Poisson recognisable in an exam question.

Inputs

The formula used

P(X = k) = e^(−λ) λ^k ÷ k!

Models the number of independent events in a fixed interval when they occur at a constant average rate. The interval must match the rate: if λ is per hour, k is a count per hour.

P(X = k)

0.132169

Mean and variance both equal λ
3.5000
Standard deviation
1.8708
P(X = 0)
0.030197
P(X ≤ k)
0.857614
P(X ≥ k)
0.274555
Most likely count
3

Mean equals variance, which is how you spot it.

The Poisson distribution has the unusual property that its mean and variance are the same number, and that is the diagnostic worth remembering. If count data has a variance much larger than its mean it is overdispersed and the Poisson is the wrong model — a negative binomial usually fits better. Exam questions signal Poisson with phrases like average number per hour and independent events.

Questions about poisson distribution

When does Poisson approximate the binomial?
When n is large and p small, with λ = np. It is much easier to compute by hand and accurate for n above about fifty with np below ten.
Can λ be non-integer?
Yes, and usually is — a rate of 3.5 arrivals an hour is perfectly ordinary. The count k, however, must be a whole number.
What breaks the Poisson assumptions?
Clustering or dependence between events. Bus arrivals bunch, so they are famously not Poisson despite being the textbook example.

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