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📊 Poisson Distribution Calculator

Calculate P(X=k), P(X≤k), and P(X>k) for a Poisson distribution. Visualise the probability mass function as a bar chart.

What is the Poisson Distribution?

The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space, when these events occur at a known average rate (λ, lambda) and independently of one another. Named after French mathematician Siméon Denis Poisson, who described it in 1837, the distribution is characterised by a single parameter λ — the mean (and also the variance) number of events in the interval. It is widely used to model count data in situations where events are rare relative to the number of possible opportunities.

The Poisson distribution is appropriate when: events occur randomly and independently (one event does not affect the probability of another); events cannot overlap (two events cannot occur at exactly the same instant); and the rate λ is constant across the observation period. Classic Poisson processes include radioactive decay counts, insurance claim arrivals, server request rates, and defect counts in manufacturing. The distribution is particularly useful when the average rate is relatively low — at high rates (λ > 30), the normal distribution provides a good approximation.

The probability mass function (PMF) gives the probability of exactly k events: P(X = k) = (e^−λ × λ^k) / k!. The cumulative distribution function (CDF) gives the probability of k or fewer events: P(X ≤ k) = Σ P(X = i) for i = 0 to k. In practice, the CDF is most commonly used for decision-making — for example, determining the probability that a call centre receives no more than 20 calls in the next hour given a mean arrival rate of 15 calls per hour.

Poisson Distribution Formulas

PMF
P(X = k) = (λᵏ × e⁻λ) / k!
Properties
Mean = λ
Variance = λ
Std Dev = √λ

Common Applications of Poisson Distribution

  1. 1
    Call Centre Modelling
    Estimating the probability of receiving exactly k calls per hour when the average rate is λ calls/hour.
  2. 2
    Quality Control
    Modelling the number of defects per unit of product when defects occur rarely and independently.
  3. 3
    Medical Events
    Predicting the number of hospital admissions per day for rare conditions.
  4. 4
    Traffic Analysis
    Modelling the number of vehicles passing a checkpoint per minute.

Frequently Asked Questions

The Poisson distribution models the number of events occurring in a fixed interval of time or space, given that events occur independently at a constant average rate λ. It is used for rare events.

Lambda is the average number of events per interval. It is both the mean and variance of the Poisson distribution. A larger λ shifts the distribution to the right and makes it more symmetric.

Use Poisson when n is large and p is small (rare events), such that np ≈ λ. The Poisson distribution approximates the binomial when n > 20 and p < 0.05.

The cumulative probability P(X ≤ k) is the sum of P(X = 0) + P(X = 1) + ... + P(X = k). It tells you the probability of observing at most k events.

Yes. The Poisson distribution is a discrete probability distribution — k must be a non-negative integer (0, 1, 2, ...). There is no upper limit on k, though probabilities become negligible after a few multiples of λ.

Real-World Applications

📞
Call Centre Staffing
Model the probability that a call centre receives more than 25 calls in a 5-minute period (given λ = 18 calls/5 min) to determine how many agents must be staffed to avoid queue overflow with 95% probability.
🏥
Hospital Emergency Admission Planning
Model the number of emergency admissions per hour at a hospital A&E department — the Poisson distribution underpins Erlang C queuing models used to staff emergency departments and set bed capacity targets.
☢️
Radioactive Decay Counting
The number of radioactive decay events detected by a Geiger counter in a fixed time interval follows a Poisson distribution — the classic physical application for which Rutherford and Geiger used the distribution in 1910.
🖥️
Network Traffic Analysis
Model the number of data packets arriving at a router per millisecond — Poisson arrival models are foundational to telecommunications capacity planning and network queuing theory.
🏭
Manufacturing Defect Control
Model the number of defects per unit area of fabric, rolled steel, or semiconductor wafer — a defect rate of λ = 0.5 defects/m² implies the probability of zero defects in a 2 m² panel is e^(−1) ≈ 36.8%.
🚗
Traffic Flow Modelling
Model the number of vehicles arriving at an intersection per minute during off-peak hours — Poisson flow assumptions underpin traffic signal timing models and highway capacity analysis for low-to-moderate flow conditions.

Common Mistakes

1
Applying Poisson when events are not independent
The Poisson distribution requires independence between events. If one event triggers or inhibits subsequent events (e.g., accident clusters, disease contagion), the Poisson assumption is violated — negative binomial or Hawkes process models are more appropriate for clustered or contagious count data.
2
Using a fixed λ when the rate is actually variable
The Poisson parameter λ must be constant over the observation window. Traffic arrival rates, call centre demand, and customer arrivals all vary by time of day — applying a single daily average λ to a specific 15-minute period produces incorrect probability estimates.
3
Confusing PMF and CDF
The PMF gives the probability of exactly k events; the CDF gives the probability of k or fewer events. For most practical questions ("what is the probability of more than 5 failures?"), you need 1 − CDF(5), not the PMF at k = 5. Confusing the two is a common and consequential error in reliability and risk analysis.
4
Not checking the Poisson approximation conditions
The Poisson distribution approximates binomial(n, p) when n is large and p is small (np < 10). For moderate p values (p > 0.1), the Poisson approximation deteriorates and the binomial distribution should be used directly.
5
Treating the mean and variance as independently known
The Poisson distribution has a single parameter λ that is simultaneously the mean and the variance. If observed data shows variance significantly different from the mean, the Poisson model is a poor fit — overdispersed count data (variance > mean) is better modelled by the negative binomial distribution.

Poisson PMF Quick Reference (selected λ values)

P(X = k) λ = 1 λ = 3 λ = 5
k = 0 0.3679 0.0498 0.0067
k = 1 0.3679 0.1494 0.0337
k = 2 0.1839 0.2240 0.0842
k = 3 0.0613 0.2240 0.1404
k = 5 0.0031 0.1008 0.1755
k = 10 0.0000 0.0008 0.0181

References

  1. Poisson, S.D. Recherches sur la probabilité des jugements. Bachelier, 1837.
  2. Feller, W. An Introduction to Probability Theory and Its Applications. Wiley, 1968.
  3. Haight, F.A. Handbook of the Poisson Distribution. Wiley, 1967.
  4. Cameron, A.C. and Trivedi, P.K. Regression Analysis of Count Data. Cambridge University Press, 1998.
  5. Ross, S.M. Introduction to Probability Models. Academic Press, 2019.

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