Poisson Distribution Calculator
Calculate Poisson probabilities — P(X=k), P(X≤k), and P(X>k) — for any average event rate (λ) and event count (k).
Poisson Distribution Calculator
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Formula: P(X=k) = (λ^k × e^(-λ)) / k!
Worked example — P(X=6) ≈ 10.42%
Formula
P(X=k) = (λ^k × e^(-λ)) / k!
The Poisson distribution gives the probability of observing exactly k independent events in a fixed interval, given that events occur at a known average rate λ. Mean and variance are both equal to λ.
Worked Examples
Example 1: Customer arrivals at a coffee shop
Problem:A coffee shop averages 4 customers every 10 minutes (λ = 4). What is the probability that exactly 6 customers arrive in the next 10 minutes?
Solution:P(X=6) = (4⁶ × e⁻⁴) / 6! = (4096 × 0.0183) / 720 ≈ 0.1042.
Result:P(X=6) ≈ 10.42%
Example 2: Rare defects in manufacturing
Problem:A factory finds an average of 1.5 defective units per 1,000 produced (λ = 1.5). What is the probability of finding 0 defects in the next batch of 1,000?
Solution:P(X=0) = (1.5⁰ × e⁻¹·⁵) / 0! = 1 × 0.2231 / 1 = 0.2231.
Result:P(X=0) ≈ 22.31% chance of a defect-free batch
Frequently Asked Questions
What kind of real-world situations does the Poisson distribution model?
The Poisson distribution models the count of independent events that occur at a known constant average rate within a fixed interval of time, distance, or space — customer arrivals at a store per hour, typos per page in a manuscript, radioactive decay events per second, server requests per minute, or potholes per mile of road. It's the natural choice whenever events happen randomly and independently, without a fixed upper limit on how many could occur.
What does lambda (λ) represent, and why does the calculator only need that one parameter?
Lambda (λ) is the average (expected) number of events in the interval you're modeling — for example, if a call center averages 3 calls per minute, λ = 3. Remarkably, the Poisson distribution is fully defined by this single parameter: its mean, variance, and standard deviation are all derived directly from λ (mean = λ, variance = λ, so standard deviation = √λ), unlike the normal distribution which needs both a mean and a standard deviation specified independently.
What is the difference between P(X=k), the PMF, and P(X≤k), the CDF, in Poisson Distribution Calculator's result?
P(X=k), the probability mass function (PMF), gives the exact probability of observing precisely k events — no more, no fewer. P(X≤k), the cumulative distribution function (CDF), gives the probability of observing k events or fewer, summing the PMF across all values from 0 up to k. Most practical questions ('what's the chance of at least/at most this many events?') require the CDF, not just the PMF.
Why is the variance of a Poisson distribution always equal to its mean?
This is a defining mathematical property of the Poisson distribution, derived directly from its formula — both the mean and variance equal λ. This unique 'mean equals variance' relationship is often used as a diagnostic check: if real-world count data has a variance much larger than its mean (called overdispersion), a Poisson model may not fit well, and a Negative Binomial distribution is often used instead.
How is the Poisson distribution used in queueing theory and staffing decisions?
Call centers, hospital emergency rooms, and web servers all use Poisson models to predict how many arrivals to expect in a given time window, which then feeds into staffing formulas (like the Erlang C formula) to determine how many staff or servers are needed to keep wait times acceptably short during that period.
Can the Poisson distribution model events that happen less than once per interval on average?
Yes — λ can be any positive number, including fractions less than 1. For example, if a website receives an average of 0.3 critical errors per day (λ = 0.3), the Poisson distribution still correctly gives the probability of exactly 0, 1, 2, or more errors occurring on any given day, even though the average itself is below one event.
What real disasters or rare-event models famously use the Poisson distribution?
The Poisson distribution's most famous historical application analyzed Prussian cavalry deaths from horse kicks per army corps per year (Ladislaus Bortkiewicz, 1898) — a genuinely rare, independent event occurring at a roughly constant rate, which matched the Poisson model closely. Modern uses include modeling earthquake occurrence rates, insurance claim counts, and radioactive decay counts in a Geiger counter.
How do I rescale lambda when my interval is longer or shorter than the one the rate was measured over?
In a Poisson process the rate parameter scales linearly with the size of the interval: if events occur at rate r per unit time, then over an interval of length t you use λ = r × t. A server averaging 12 requests per minute has λ = 12 for one minute, λ = 3 for a 15-second window, and λ = 720 for an hour. Always rescale λ before entering it, because k must be counted over the same interval.
How does the Poisson distribution relate to the exponential distribution?
They describe the same underlying Poisson process from two angles: Poisson counts how many events occur in a fixed interval, while the exponential distribution describes the waiting time between consecutive events, with mean 1/λ. If arrivals are Poisson at 4 per hour, the average gap between arrivals is 0.25 hours. The exponential waiting time is memoryless, which is why time already spent waiting does not change the expected remaining wait.
What is the normal distribution and why does it appear so often?
The normal (Gaussian) distribution is a bell-shaped curve defined by its mean and standard deviation. About 68% of data falls within 1 SD, 95% within 2 SD, and 99.7% within 3 SD of the mean. The Central Limit Theorem explains its prevalence: sample means approach normality regardless of the underlying distribution.
References
Reviewed for accuracy by Manoj Kumar, Mathematics Educator · Editorial policy
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