Poisson Distribution Calculator

The Poisson distribution is the standard tool for counting independent events that happen at a known average rate, whether that rate is calls per minute in a call centre, decay events per second from a radioactive sample, accidents per day on a stretch of road, insurance claims per quarter, web requests per second on a server, machine failures per shift, or typos per printed page. It was introduced by the French mathematician Siméon-Denis Poisson in his 1837 treatise on the probability of judgements and convictions, and the equation P(X = k) = λᵏ · e^(−λ) / k! is the same one statisticians still use today. The calculator below computes the probability mass function (PMF), the cumulative distribution function (CDF), the inverse CDF (quantile), and the running moments for any rate λ greater than zero, so a single screen replaces a table look-up and a calculator.

For a count variable the Poisson has the rare and elegant property that mean and variance are both equal to λ, and the standard deviation is √λ. This equal-mean-and-variance signature is also a quick visual check: empirical data that show variance visibly bigger than the mean is over-dispersed and not Poisson; data with variance much smaller than the mean is under-dispersed and not Poisson either. Telecom traffic, hospital arrivals, and radioactive decay are good fits; rainfall totals and stock returns are not.

How to Use the Poisson Distribution Calculator

1. Enter the rate parameter λ in the first input. This is the average number of events per unit of time, area, or volume. The default is three events per unit; anything strictly greater than zero is accepted.
2. Pick a mode with the radio buttons:
   • PMF, find P(X = k) for an exact count, useful for "what is the chance of exactly k events in one interval?"
   • CDF, find P(X ≤ k) for an upper bound on the count, useful for sizing capacity or testing one-sided hypotheses.
   • Quantile, smallest k with P(X ≤ k) ≥ p for the reverse lookup, useful when you need the count that covers a given percentile.
3. In the count input, type the non-negative integer k for the PMF and CDF modes. Decimals are floored because Poisson is on the integers.
4. For the quantile mode, type the target probability p in (0, 1] instead. The calculator returns the smallest integer k whose cumulative probability just reaches or crosses p.
5. Press Calculate. The result box prints the requested probability along with the live mean, variance, standard deviation, and a plain-English reading of the answer. Seven worked examples under the inputs let you verify the result against reference values from scipy.stats.poisson.

The same control recycles for every mode, so switching between PMF, CDF, and quantile does not require clearing the inputs. Common values λ = 2 with k = 0, λ = 3 with k = 2, λ = 10 with k = 10, λ = 0.5 with k = 0, and λ = 4 with k = 4 are provided as buttons for fast verification.

The Formula

The Poisson probability mass function is

P(X = k) = λᵏ · e^(−λ) / k!

for k = 0, 1, 2, … and rate λ greater than zero. The cumulative distribution function is the running sum:

P(X ≤ k) = Σᵢ₌₀ᵏ λⁱ · e^(−λ) / i!

The mean (expected value), variance, and standard deviation all collapse into simple expressions in λ:

• Mean: E[X] = λ
• Variance: Var[X] = λ
• Standard deviation: σ = √λ

For very large k the formula λᵏ / k! overflows a normal floating-point register; the calculator therefore evaluates the PMF in log-space using the Lanczos approximation for the log-gamma function, and exponentiates only at the end. The CDF is summed term-by-term in the same log-space, with an early-exit when a term becomes negligibly small compared with the running sum. The quantile walks the CDF forward from k = 0 until it crosses the target probability, which is the standard inverse-CDF algorithm for discrete distributions. The result is a calculator that stays accurate for λ up to a few hundred and k up to several thousand without any user tuning.

Worked Examples

Example 1, Quiet hour at a help desk. Calls arrive at an average rate λ = 2 per hour. The probability that the next hour sees zero calls is

P(X = 0) = 2⁰ · e^(−2) / 0! = 1 · 0.1353 / 1 = 0.1353

so about 13.5% of quiet hours stay completely silent. Plugging λ = 2 and k = 0 into the calculator returns 0.13534, matching the scipy reference to four decimals.

Example 2, Patients arriving at urgent care. Arrivals average λ = 3 per hour at a walk-in clinic. The probability that exactly two patients show up in a given hour is

P(X = 2) = 3² · e^(−3) / 2! = 9 · 0.049787 / 2 = 0.2240

so about 22.4% of hours bring exactly two patients. The calculator confirms 0.22404 for λ = 3, k = 2.

Example 3, A busier service point. Front-door arrivals average λ = 10 per hour. The probability that ten arrive in a given hour is

P(X = 10) = 10¹⁰ · e^(−10) / 10! = 0.1251

so a peak hour of exactly ten arrivals is more than one-in-eight, the Poisson distribution is centred on λ for moderate rates, so k = λ is the most likely count. Calculator output for λ = 10, k = 10 is 0.12511.

Example 4, Rare event with very small λ. With rate λ = 0.5 per minute, the chance that no event occurs in a minute is

P(X = 0) = 0.5⁰ · e^(−0.5) / 0! = e^(−0.5) = 0.6065

The probability is high because the rate is well under one event per minute. Calculator output for λ = 0.5, k = 0 is 0.60653.

Example 5, Right at the mode for λ = 4. When the rate is four per unit interval, the most likely count is exactly four. The probability is

P(X = 4) = 4⁴ · e^(−4) / 4! = 256 · 0.018316 / 24 = 0.1954

so about 19.5% of intervals see exactly four events. Calculator output for λ = 4, k = 4 is 0.19537.

Example 6, Cumulative upper bound. With λ = 2, the probability that at most three events occur is

P(X ≤ 3) = P(0) + P(1) + P(2) + P(3) = 0.1353 + 0.2707 + 0.2707 + 0.1804 = 0.8571

Calculator output for λ = 2 in CDF mode with k = 3 returns 0.85712; the complementary tail P(X > 3) = 1 − 0.85712 = 0.14288 is the chance of four or more events in the interval. This is the typical size question, "how often do we get four or more customers in an hour?"

Example 7, Quantile for capacity planning. With λ = 3 arrivals per hour, what is the smallest k for which P(X ≤ k) ≥ 0.80? Compute the running CDF: P(X ≤ 3) = 0.6472, P(X ≤ 4) = 0.8153. The smallest qualifying k is 4. Switch the calculator to quantile mode, enter p = 0.8, and read k = 4, meaning that provisioning for four arrivals per hour covers 80% of intervals.

Where It Shows Up

The Poisson distribution is the natural model anywhere events arrive independently at a roughly constant average rate. The same λᵏ · e^(−λ) / k! expression appears in:

• Call-centre and contact-centre arrivals. Erlang-style queueing models lean on Poisson arrivals for staffing problems; Erlang-C and Erlang-B are direct generalisations. The arrival rate λ becomes the foundation for both service-level equations and minimum-agent calculations.
• Radioactive decay. The half-life of a sample, the time between decay events, is exponential with mean 1/λ; the count of decays in a fixed interval is Poisson with rate proportional to interval length. This relationship is what makes carbon-14 dating and most nuclear-medicine imaging work.
• Traffic accidents per day or per kilometre. Road-safety statistics treat accident counts as Poisson over a fixed segment and a fixed time window, so λ becomes the design basis for guard-rail placement and accident-reduction targets.
• Insurance claims per policy per quarter. Actuarial loss models start from a Poisson frequency and then attach a severity distribution to each claim. Premiums in property-and-casualty, health, and warranty insurance all decompose this way.
• Web requests per second on a server. Load-testing tools and capacity-planning models assume Poisson-incoming traffic to compute the queue length and the probability of dropping a request. Real traffic shows occasional bursts, so the model is an upper bound for stable workloads.
• Machine failures per shift. Reliability engineering counts field failures as Poisson events with rate proportional to fleet size, then extracts mean-time-between-failures, MTBF, as 1/λ.
• Typos per page in a manuscript. Book and report editors count misspellings per page as roughly Poisson for quality-control sampling, λ being the author's error rate.

The unifying feature is that the data are non-negative integers that can be counted in a fixed interval.

Common Mistakes

Even a simple distribution has pitfalls. Five mistakes come up most often:

• Setting λ ≤ 0. The rate parameter is strictly positive by construction. A negative rate is not a meaningful value in any real-world situation the Poisson covers. The calculator rejects it rather than silently returning nonsense.
• Computing λᵏ / k! directly for large k. For λ = 100 and k = 100, naïve evaluation overflows 64-bit floats long before reaching the answer. The calculator evaluates log P(X = k) and exponentiates only at the end, an essential step in any production implementation.
• Confusing PMF and CDF. P(X = k) is the probability of exactly k events; P(X ≤ k) is the cumulative probability of zero through k. Both are useful, but they answer different questions. Queueing capacity uses CDF, anomaly detection usually uses PMF.
• Using Poisson as a substitute for a small-p binomial. The Poisson approximation to the binomial is excellent when n is large and p is small, but it can bias results when n·p is below about 10. For very small n and large p, compute the binomial directly.
• Assuming independent arrivals. The Poisson derivation assumes events are independent, which is what makes the elegant merge of arrivals, queueing, and renewal processes work. If arrivals cluster (earthquake aftershocks, network retransmits, customer referrals), the Poisson underestimates the tails and a negative-binomial or compound Poisson model fits better.

Frequently Asked Questions

What is the Poisson distribution in plain English? It is the probability distribution of the count of independent events that happen at a known average rate. If events happen at an average rate of λ per unit time, the chance of seeing exactly k events in one unit is λᵏ · e^(−λ) / k!. The same rule covers any "count events in a fixed window" problem, from radioactive decay to call-centre arrivals.

Who introduced the Poisson distribution? The French mathematician Siméon-Denis Poisson introduced it in his 1837 work Recherches sur la probabilité des jugements en matière criminelle et en matière civile. The distribution was originally intended to model the number of wrongful convictions across trials; its applications quickly spread to anything that counts independent events at a constant average rate.

When can I use the Poisson distribution? Use it when four conditions hold: events are countable as non-negative integers, they occur independently, the average rate λ is constant over the interval of interest, and two events cannot occur at the exact same instant. Under these assumptions, the Poisson is the unique distribution for the count.

What is the difference between Poisson and binomial? The binomial counts successes in a fixed number of trials; the Poisson counts events in a fixed interval. The Poisson is also the limit of the binomial when n is large and p is small but their product n · p = λ is fixed. They look alike for small probabilities, but the Poisson is the better model whenever the underlying process has no upper bound on the count.

How is the Poisson used in queueing theory? Queueing theory, especially the M/M/1 and M/M/c Erlang models, assumes Poisson arrivals. The arrival rate λ becomes the foundation for service-level equations, average wait, and the probability that a queue exceeds a given length. The Erlang B and Erlang C formulas are direct consequences.

Can the Poisson distribution model over-dispersed data? Not directly. Over-dispersed data have variance visibly larger than mean; the Poisson forces mean = variance exactly. For over-dispersed counts, the negative-binomial distribution adds a dispersion parameter and is the standard extension. For under-dispersed counts (variance much smaller than mean), the binomial with finite N is more appropriate.

What is the simplest way to sample from a Poisson? For very small λ, the inverse-CDF method is fine. For larger λ the standard Knuth algorithm, generate uniform u, set k = 0, multiply by λ/(k+1), repeat while the running product exceeds u, return k, runs in expected O(λ) time and is the textbook approach. For λ above about 30 the rejection method of Atkinson or the converted algorithm of Hörmann is faster.

How accurate is this Poisson calculator? Probabilities match scipy.stats.poisson to at least five decimals for the reference cases λ = 2, 3, 10, 0.5, 4 with k = 0, 2, 10, 0, 4 respectively. For very large λ or k the relative error remains well below 1 × 10⁻⁵ because the calculation is done in log-space with the Lanczos log-gamma approximation.

How is the Poisson calculator different from a binomial calculator? The Poisson has a single parameter λ, while the binomial has two (n trials and p success probability per trial). They overlap for small p and large n, but Poisson is the right choice whenever there is no fixed upper bound on the count, as is the case for queueing, decay, and most arrivals data.

Is the Poisson distribution appropriate for rare events? Yes, and this is its original motivation. In the rare-events limit (n → ∞ and p → 0 with n · p → λ) the binomial converges to the Poisson, which is why the Poisson turns up in models of rare accidents, rare word occurrences, rare mutations, and rare system failures.

How does the Poisson distribution relate to the exponential distribution? The exponential distribution models the waiting time between Poisson events. If events arrive at rate λ, the inter-arrival time is exponential with mean 1/λ. Together they form the classic Poisson process, which is the building block of Erlang queueing models and continuous-time Markov chains.

What is the quantile (inverse CDF) of the Poisson distribution? The quantile at probability p is the smallest non-negative integer k such that P(X ≤ k) ≥ p. The calculator implements this by walking the CDF forward from k = 0 until the running sum reaches p. It is used when one needs to size capacity for a target percentile, such as "how many arrivals will happen in 95% of hours?"

References

• Poisson, S.-D. (1837). Recherches sur la probabilité des jugements en matière criminelle et en matière civile. Paris: Bachelier.
• Feller, W. (1968). An Introduction to Probability Theory and Its Applications, Vol. I, 3rd ed. Wiley. Chapter VI is the standard textbook treatment.
• NIST/SEMATECH e-Handbook of Statistical Methods. Section 7.3.1, Poisson distribution, accessed via the NIST Engineering Statistics Handbook.
• scipy.stats.poisson documentation, scipy.stats.poisson.pmf, cdf, and ppf reference implementation, used to verify the calculator's numerical outputs.
• Haight, F. A. (1967). Handbook of the Poisson Distribution. Wiley. Comprehensive applied reference for traffic, queueing, and rare-event modelling.