0.84224337
5
5
2.2361
0.84224337
5
5
2.2361
The Poisson Distribution Calculator computes the probability mass function (PMF), mean, variance, and standard deviation for the Poisson distribution. Named after French mathematician Simeon Denis Poisson who published it in 1837, this distribution models the number of events occurring in a fixed interval of time or space, given a known average rate and independent occurrences. It is one of the most important discrete distributions in applied probability and statistics.
The Poisson distribution is the go-to model for counting rare, independent events over a continuous domain. It answers questions like: How many customers will arrive at a store in the next hour? How many defects will be found on a manufactured sheet of glass? How many calls will a call center receive per minute? How many mutations occur in a stretch of DNA? How many meteorites strike the Earth per year? How many typos appear on a page of text? In each case, events occur at some average rate λ (lambda), but the exact number in any given interval is random.
The key property of the Poisson distribution is that its mean and variance are both equal to λ. This unique characteristic, known as equidispersion, makes it easy to identify Poisson-distributed data: if the sample mean and variance are approximately equal, the Poisson model may be appropriate. When the observed variance significantly exceeds the mean (overdispersion), alternative distributions such as the negative binomial should be considered. Conversely, if the variance is less than the mean (underdispersion), models like the Conway-Maxwell-Poisson may be more suitable.
The Poisson distribution has a single parameter λ > 0, which represents both the expected number of events and the variance. As λ increases, the distribution becomes more symmetric and increasingly resembles a normal distribution with μ = σ² = λ, which follows from the Central Limit Theorem. For small λ (less than about 5), the distribution is noticeably right-skewed with most probability mass concentrated near zero, while for large λ the bell-curve shape becomes evident.
This calculator uses a log-space computation for the PMF to avoid numerical overflow from large factorials. The formula ln(PMF) = -λ + k·ln(λ) - ln(k!) is computed first, then exponentiated. This approach accurately handles k values up to 50 and λ values up to 100. The Poisson distribution is indispensable in queueing theory (Erlang formulas for call center staffing), insurance claim modeling (aggregate loss distributions), epidemiology (disease incidence rates), telecommunications (packet arrivals in networks), ecology (species abundance counts), and reliability engineering (component failure prediction). Whether you are analyzing server request rates, modeling radioactive decay events, or estimating the expected number of accidents at an intersection, this calculator delivers precise probabilities instantly.
The Poisson PMF gives the probability of observing exactly k events when the expected rate is λ:
$$P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$$
To avoid overflow, we compute in log-space:
$$\ln P(X = k) = -\lambda + k \ln(\lambda) - \ln(k!)$$
then P(X = k) = exp(ln P). The mean, variance, and mode of the Poisson distribution are:
$$E[X] = \lambda, \quad \text{Var}(X) = \lambda, \quad \text{Mode} = \lfloor \lambda \rfloor$$
The standard deviation is σ = √λ. The Poisson distribution can be derived as the limit of the binomial distribution when n → ∞ and p → 0 such that np = λ remains constant.
The PMF P(X = k) gives the probability of observing exactly k events in the given interval. For example, if λ = 5 and PMF(3) = 0.1404, there is a 14.04% chance of exactly 3 events occurring. The mean λ is the expected number of events per interval. The variance equals λ (a defining property of the Poisson distribution). If your observed variance is much larger than the mean, the data may be overdispersed and better modeled by a negative binomial distribution.
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If a call center receives an average of 5 calls per hour, the probability of receiving exactly 3 calls in a given hour is about 14.04%. The standard deviation of 2.24 indicates moderate variability around the mean.
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With an average of 2 defects per unit, the probability of finding zero defects is about 13.53%. This is e^(-2) ≈ 0.1353, a quick way to compute P(X=0) for any Poisson distribution.
Use the Poisson distribution when: (1) you are counting the number of events in a fixed interval (time, area, volume), (2) events occur independently, (3) the average rate λ is constant, and (4) two events cannot occur at exactly the same instant. Common applications include modeling arrivals, defects, accidents, mutations, and natural disasters.
This is a mathematical property derived from the moment-generating function of the Poisson distribution. Both E[X] and Var(X) equal λ. In practice, this property helps identify Poisson-distributed data: compute the sample mean and variance, and if they are approximately equal, the Poisson model is plausible. This property is called equidispersion.
The Poisson distribution is the limiting case of the binomial distribution when n is large, p is small, and np = λ is moderate. Specifically, as n → ∞ and p → 0 with np → λ, Binomial(n, p) → Poisson(λ). A rule of thumb: the Poisson approximation is good when n ≥ 20 and p ≤ 0.05.
Yes, λ can be any positive real number. For example, λ = 3.7 means an average of 3.7 events per interval. While the observed count k must be a non-negative integer, the rate parameter λ is continuous. Common examples include λ = 0.5 (rare events) or λ = 47.3 (frequent events).
If your data shows variance significantly greater than the mean, this is called overdispersion, and the Poisson model is inadequate. Common alternatives include the negative binomial distribution (which adds an extra parameter for overdispersion), quasi-Poisson models, or zero-inflated Poisson models (if excess zeros are the issue).
For large λ (typically λ ≥ 20), the Poisson distribution is well-approximated by a normal distribution with mean μ = λ and variance σ² = λ. This follows from the Central Limit Theorem, since a Poisson(λ) random variable can be viewed as the sum of many independent Poisson(λ/n) variables.
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