🎲 Poisson Distribution Calculator – PMF, CDF & Probability
The Poisson distribution is one of the most important discrete probability distributions in statistics, engineering, and the natural sciences. It models the number of times a random event occurs in a fixed interval of time or space — provided those events happen independently at a known, constant average rate λ (lambda). This calculator computes exact, cumulative, upper-tail, and interval Poisson probabilities instantly, and visualises the full distribution with a PMF bar chart and probability table.
🔢 The Poisson Formula (PMF)
The probability of observing exactly k events when the average rate is λ is given by the Probability Mass Function (PMF):
P(X = k) = e^(−λ) × λ^k / k!where e ≈ 2.71828 is Euler's number and k! = k × (k−1) × … × 1 is the factorial of k. This calculator evaluates the formula in log-space (using log-gamma) to remain numerically stable for large values of k or λ, avoiding floating-point overflow that would affect naïve implementations.
| Mode | Formula | Use When… |
|---|---|---|
| Exact P(X = k) | e^(−λ) × λ^k / k! | You need the probability of precisely k events |
| Cumulative P(X ≤ k) | Σᵢ₌₀ᵏ P(X = i) | You need the probability of at most k events |
| Upper Tail P(X ≥ k) | 1 − P(X ≤ k − 1) | You need the probability of at least k events |
| Interval P(a ≤ X ≤ b) | P(X ≤ b) − P(X ≤ a − 1) | You need a bounded event-count range |
📐 Calculation Modes Explained
This tool supports five modes to cover every common probability question:
- Exact P(X = k) — the probability of observing precisely k events. Useful for spotting the most likely count or evaluating a specific scenario.
- Cumulative P(X ≤ k) — the probability of at most k events; sums all PMF values from 0 to k. Use this for "no more than" questions in quality control or risk modelling.
- Upper Tail P(X ≥ k) — the probability of at least k events; equals 1 − P(X ≤ k − 1). Use this for "at least" questions such as service-level thresholds.
- Interval P(a ≤ X ≤ b) — the probability the count falls within a bounded range; equals P(X ≤ b) − P(X ≤ a − 1).
- Distribution Summary — generates a full PMF/CDF table over a custom k range, with mean, variance, and standard deviation.
📊 Distribution Statistics
The Poisson distribution has a remarkable property — its mean and variance are both equal to λ:
Mean (μ)
μ = λ
Expected number of events
Variance (σ²)
σ² = λ
Spread equals the rate
Std Dev (σ)
σ = √λ
Square root of the rate
Skewness = 1 / √λ and excess kurtosis = 1 / λ — as λ grows, the Poisson distribution becomes more symmetric and approaches a normal distribution shape.
✅ When to Use the Poisson Distribution
The Poisson model is appropriate when four conditions hold:
- Independence — one event occurring does not affect whether another occurs.
- Constant rate — the average rate λ is constant throughout the interval.
- Rare simultaneous events — two events cannot occur at the exact same instant.
- Discrete counts — you are counting occurrences, not measuring a continuous quantity.
Common real-world applications include: call centre arrivals, website requests per second, manufacturing defects per batch, insurance claims per day, and radioactive decay events per minute.
🔄 Poisson Approximation to the Binomial
When a binomial experiment has a large n (≥ 20) and a small p (≤ 0.05), the Poisson distribution with λ = np provides a fast and accurate approximation. For example, 500 items inspected with a 0.2% defect rate (n = 500, p = 0.002) gives λ = 1.0, and the Poisson model closely matches the exact binomial probabilities.
⚠️ Limitations and Edge Cases
If your data shows overdispersion (variance significantly greater than the mean), a Poisson model may underfit — a negative binomial distribution is often a better choice. For very small probabilities at large k, results may round to zero due to floating-point limits, though this only affects negligibly small probabilities that have no practical significance.