0
0.265026
5
2.2361
0
0.265026
5
2.2361
The Poisson Probability Calculator computes exact and cumulative probabilities for the Poisson distribution — the model for counting the number of events that occur in a fixed interval of time, space, or other continuous measure when events occur independently at a constant average rate. Named after French mathematician Siméon Denis Poisson, this distribution is ubiquitous in science, engineering, and business.
Applications include modeling customer arrivals, radioactive decay events, website hits, insurance claims, manufacturing defects, and any process where events occur randomly but at a known average rate. Enter the average rate (λ) and desired event count (k) to compute instant probabilities.
The Poisson probability mass function gives the probability of exactly k events:
$$P(X = k) = \frac{e^{-\lambda} \lambda^k}{k!}$$
Where:
The cumulative probability sums the PMF from 0 to k:
$$P(X \leq k) = \sum_{i=0}^{k} \frac{e^{-\lambda} \lambda^i}{i!}$$
A unique property of the Poisson distribution is that both its mean and variance equal λ:
$$E(X) = \lambda \quad ; \quad \text{Var}(X) = \lambda \quad ; \quad \sigma = \sqrt{\lambda}$$
This calculator uses logarithmic computation for numerical stability: ln P(X=k) = -λ + k·ln(λ) - ln(k!), where ln(k!) is computed via the Stirling log-gamma approximation. The CDF is computed using the recursive relationship P(X=i) = P(X=i-1) × λ/i, starting from P(X=0) = e^(-λ).
The Poisson model requires: (1) events occur independently; (2) the average rate is constant; (3) two events cannot occur at exactly the same instant; and (4) the probability of an event in a tiny interval is proportional to the interval length. These conditions define a Poisson process, one of the most fundamental stochastic models.
P(X = k) tells the probability of exactly k events occurring. If a call center receives λ = 10 calls per hour, P(X = 15) ≈ 3.47% indicates that receiving exactly 15 calls in an hour is relatively uncommon but not rare. P(X ≤ k) helps with capacity planning: P(X ≤ 12) gives the probability that 12 or fewer calls arrive, informing staffing decisions.
The standard deviation √λ indicates the typical spread of event counts around the mean. For λ = 25, σ = 5, so most observations will fall between 20 and 30 events (within one standard deviation of the mean).
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With an average rate of 5 calls per hour, P(exactly 3 calls) ≈ 14.04%. P(3 or fewer calls) ≈ 26.5%, meaning there's about a 1 in 4 chance of a slow hour with 3 or fewer calls. This information helps plan minimum staffing levels.
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A website averages 7 errors per day. P(exactly 10 errors) ≈ 7.1%. P(10 or fewer) ≈ 90.1%, so exceeding 10 errors happens about 10% of the time. An alert threshold at 10+ errors would trigger roughly once every 10 days.
Use the binomial when you have a fixed number of trials with a known success probability. Use the Poisson when counting events in a continuous interval (time, area, volume) with a known average rate. The Poisson is the limit of the binomial as n → ∞ and p → 0 with np = λ. Rule of thumb: if n > 20 and p < 0.05, the Poisson approximation B(n, p) ≈ Poisson(np) is accurate and simpler to compute.
The equidispersion property (mean = variance = λ) is a defining characteristic of the Poisson distribution. In real data, if the sample variance significantly exceeds the sample mean (overdispersion), the data likely does not follow a Poisson distribution — consider the negative binomial distribution instead. If the variance is less than the mean (underdispersion), consider the binomial or Conway-Maxwell-Poisson distribution.
Yes. λ represents an average rate and can be any positive real number. For example, a store might average 3.7 customers per hour. The observed count k must be a non-negative integer (you can't have 3.7 customers arrive), but the rate λ can be any positive value. Fractional λ values simply mean the average falls between two whole numbers.
A Poisson process is a continuous-time stochastic process where events occur randomly and independently at a constant rate λ. Key properties: (1) the number of events in any interval follows a Poisson distribution; (2) the time between consecutive events follows an exponential distribution with mean 1/λ; (3) events in non-overlapping intervals are independent. It is the simplest model for random event arrivals and serves as the building block for more complex queueing and reliability models.
If events follow a Poisson process with rate λ, the waiting time between consecutive events follows an exponential distribution with parameter λ (mean waiting time = 1/λ). If a bus arrives on average every 10 minutes (λ = 6 per hour), the time until the next bus is Exponential(6), with mean 10 minutes. The Poisson counts events in intervals; the exponential measures intervals between events. They are two perspectives on the same process.
The Poisson model assumes: constant rate (no time-of-day effects), independence (no clustering or contagion), and impossibility of simultaneous events. Real-world violations include: varying arrival rates (use non-homogeneous Poisson), clustered events like earthquakes (use Hawkes process), overdispersed counts (use negative binomial), and zero-inflated data where zeros are more common than expected (use zero-inflated Poisson). Always check if the equidispersion property holds before applying the Poisson model.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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