Inverse Poisson Calculator

The inverse Poisson distribution, or Poisson quantile function, turns a cumulative probability back into a count. If X counts random events with mean λ, the quantile Q(p) is the smallest k such that P(X ≤ k) ≥ p.

The Poisson distribution models events that happen independently at a steady average rate: calls per hour, typos per page, radioactive decays per second, or goals per match. Its probabilities are

P(X = k) = e^−λ·λ^k / k!

1. Mean and variance are both λ, so the standard deviation is √λ.
2. The shape is skewed right for small λ and becomes bell-shaped as λ grows.
3. The median is close to λ + 1/3 − 0.02/λ.

Add Poisson probabilities for k = 0, 1, 2, … and stop at the first k where the running total reaches p.

1. Start at zero: P(X = 0) = e^−λ. For λ = 4, that is 0.0183.
2. Use the recursion: P(X = k) = P(X = k − 1) × λ / k. Each new probability comes from the last one with a single multiplication.
3. Keep a running total of the probabilities, which is the CDF.
4. Stop at the first k where the total is ≥ p. For λ = 4 and p = 0.9, the total passes 0.9 at k = 7 (0.9489).

For an upper-tail critical value, find the smallest k with 1 − P(X ≤ k − 1) ≤ p instead.

For large λ the Poisson distribution looks like a normal curve with mean λ and standard deviation √λ, so k ≈ λ + z·√λ − 0.5. The −0.5 is a continuity correction that accounts for whole-number counts.

The approximation is poor for small λ because the Poisson shape is skewed: for λ = 2 and p = 0.95 it suggests k ≈ 3.8, but the exact answer is 5. By λ = 50 the two usually agree to within one count.

A more accurate shortcut uses the square-root transform: √X is close to normal with mean √λ and standard deviation ½, so k ≈ (√λ + z/2)².

Inverting the Poisson CDF in λ gives an exact confidence interval for the true rate from one observed count. This is the Garwood interval, and it is what inverse Poisson calculations are most often used for in medicine, reliability, and physics.

• Lower limit: the λ where P(X ≥ k) = α/2. Below it, seeing k or more would be too unlikely.
• Upper limit: the λ where P(X ≤ k) = α/2. Above it, seeing k or fewer would be too unlikely.
• Zero events: the lower limit is 0 and the one-sided 95% upper limit is −ln(0.05) ≈ 3.0.

Divide both limits by the exposure (person-years, hours, square metres) to turn them into a rate interval.

When demand arrives randomly at an average rate λ, the p-quantile is the capacity that covers a fraction p of periods.

• Inventory: a shop sells 6 units of an item per day on average. To avoid running out on 95% of days it must start each day with qpois(0.95, 6) = 10 units.
• Call centers: staffing for the 90th percentile of hourly calls keeps queues short most of the time.
• Hospital beds and ICU capacity are planned from the quantiles of daily admissions.
• Quality control: the upper critical value sets the alarm threshold for defect counts.

Raising the service level from 95% to 99% often costs only two or three extra units, which the simulation lets you see.