Poisson Distribution Calculator
Enter the average event rate (λ) and desired number of events (k) to get P(X=k), P(X≤k), and P(X≥k) — with mean, variance, and step-by-step Poisson probability calculation.
Enter your values above to see the results.
Tips & Notes
- ✓Lambda (λ) must match the interval you are analyzing. If events occur at 2 per hour and you want probabilities for a 30-minute window, use λ=1 (not 2). Always scale λ to match your time or space interval.
- ✓Mean = Variance = λ in the Poisson distribution. If your data shows mean ≠ variance, the Poisson model may be wrong — overdispersion (variance > mean) is common and requires a negative binomial distribution instead.
- ✓For large λ (>20), the Poisson distribution approximates a normal distribution with μ=λ and σ=√λ. You can use normal approximation to simplify calculations: P(X ≤ k) ≈ Φ((k+0.5−λ)/√λ).
- ✓P(X ≥ k) = 1 − P(X ≤ k−1), not 1 − P(X ≤ k). For "at least 5 events": use 1 minus the cumulative probability up to k=4, not k=5.
- ✓The Poisson distribution only accepts non-negative integer values for k (0, 1, 2, 3, ...). The rate λ can be any positive real number, including non-integers like λ=2.5 or λ=0.3.
Common Mistakes
- ✗Using the wrong time unit for λ. If the average rate is 12 calls per hour and you want probabilities for a 5-minute window, use λ=12×(5/60)=1. Applying λ=12 to a 5-minute interval is a 12× error in the rate.
- ✗Applying Poisson when events are not independent. Disease transmission, social contagion, and clustered events violate independence. One infection triggers more infections — the rate is not constant. Use a different distribution (negative binomial, zero-inflated Poisson) for clustered counts.
- ✗Confusing P(X=k) with P(X≤k). P(exactly 3 events) = P(X=3) is a single term. P(3 or fewer events) = P(X≤3) = P(0)+P(1)+P(2)+P(3). The cumulative is always larger than the exact probability for k > 0.
- ✗Using Poisson for high rates. When λ>20, many texts suggest the normal approximation. But the Poisson distribution itself remains valid for any λ — the approximation only simplifies calculation. Use exact Poisson for any λ when precision matters.
- ✗Treating k as a continuous value. k must be a non-negative integer: 0, 1, 2, 3... P(X=2.5) is undefined in the Poisson distribution. For non-integer k, use the lower integer and compute P(X ≤ k) = P(X ≤ 2).
Poisson Distribution Calculator Overview
The Poisson distribution models the number of times a rare event occurs in a fixed interval of time, space, or volume — when events happen independently and at a known average rate. Customer arrivals per hour, server requests per second, defects per unit area, accidents per month — all follow the Poisson distribution when the rate is constant and events don't cluster or repel each other.
Poisson probability — exactly k events:
P(X = k) = (e⁻λ × λᵏ) / k!
EX: A call center receives 3 calls per minute on average (λ=3). P(exactly 5 calls in one minute): P(X=5) = (e⁻³ × 3⁵) / 5! = (0.0498 × 243) / 120 = 12.09/120 = 0.1008 (10.08%)Cumulative probability — k or fewer events:
P(X ≤ k) = Σ P(X = i) for i = 0 to k
EX: P(5 or fewer calls) = P(0)+P(1)+P(2)+P(3)+P(4)+P(5) = 0.0498+0.1494+0.2240+0.2240+0.1680+0.1008 = 0.9160 (91.6%)Mean and variance of Poisson — both equal λ:
μ = λ | σ² = λ | σ = √λ
EX: λ=3 calls/min → Mean=3, Variance=3, SD=√3≈1.73 → Expect 3 calls with typical variation of ±1.73 callsPoisson conditions — when to use this distribution:
| Condition | Meaning | Example That Fails |
|---|---|---|
| Events are rare relative to opportunities | Many chances, few occurrences | 60% defect rate — not rare |
| Events are independent | One event doesn't affect others | Contagious disease — infections cluster |
| Rate is constant (λ fixed) | Average rate doesn't change | Rush-hour calls — rate varies by hour |
| Cannot have two events simultaneously | Events occur one at a time | Usually satisfied for point events |
Frequently Asked Questions
The Poisson distribution models counts of independent events occurring at a constant average rate in a fixed interval. Use it when: you count occurrences (not fractions), events are independent (one doesn't cause another), and the rate λ is constant. Examples: customers arriving per hour, server errors per day, accidents per month, typos per page. It fails when events cluster (diseases spreading) or when the rate varies over time.
Lambda is the average number of events expected in one interval — the rate parameter. λ=3 means on average 3 events occur per interval. The interval must match your question: if the rate is 6 per hour and you want probabilities for a 30-minute period, use λ=3. λ can be any positive number, including decimals. Both the mean and variance of the Poisson distribution equal λ.
Binomial counts successes in n fixed trials, each with probability p. Poisson counts events in a continuous interval with average rate λ. Binomial needs both n and p; Poisson needs only λ. When n is large (>100) and p is small (<0.01), binomial approximates Poisson with λ=n×p. Example: defect rate 0.001 in 5,000 parts → Binomial(5000, 0.001) ≈ Poisson(λ=5).
P(X=0) = e⁻λ — the probability of observing zero events in one interval. For λ=3 (e.g., 3 calls per minute): P(0) = e⁻³ = 0.0498 — only a 5% chance of silence. For λ=1: P(0) = 0.3679 → 37% chance. For λ=0.5: P(0) = 0.6065 → 61% chance. When λ<1, zero events is the most probable outcome. As λ grows, P(0) shrinks exponentially: λ=5 → P(0)=0.0067; λ=10 → P(0)=0.000045. High-rate processes almost never produce zero events.
P(X ≥ 1) = 1 − P(X=0) = 1 − e⁻λ. Example: with λ=2 events per hour, P(at least one event) = 1 − e⁻² = 1 − 0.1353 = 0.8647 (86.47%). This complement approach is always simpler than summing P(1)+P(2)+P(3)+... P(X ≥ k) for k>1: use 1 − P(X ≤ k−1), which equals 1 minus the cumulative sum up to k−1.
Yes. Yes — Poisson applies to any fixed interval: time, area, volume, or length, as long as events are independent and the rate is constant per unit. Examples: bacteria per cm² of culture plate (λ=bacteria density × plate area), defects per m² of fabric, particles per cm³ of solution. Always scale λ to your interval. Rate = 5 bacteria/cm², examining 3 cm² area → λ = 5×3 = 15 bacteria expected. P(exactly 12 in that area) = e⁻¹⁵×15¹²/12! ≈ 0.0829.