Poisson Distribution Calculator
Calculate Poisson probabilities for event counts over a fixed time, space, or exposure interval. Use this student-friendly calculator for exact, less-than, greater-than, between-values, outside-values, and cumulative Poisson distribution problems with a shaded bar chart, formula, probability table, and steps.
Background
The Poisson distribution models how many times an event occurs in a fixed interval when events happen independently and at an average rate λ. It is often used for calls per hour, emails per minute, defects per unit, arrivals per day, accidents per month, and other count-based situations.
How to use this calculator
- Enter the average rate λ for the fixed interval.
- Choose whether you want an exact, cumulative, between-values, or outside-values probability.
- Enter the event count x, or a lower and upper count for range questions.
- Click Calculate to see the probability, chart, formula, statistics, table, and steps.
- Use quick picks to explore common Poisson distribution homework examples instantly.
How this calculator works
- It uses the Poisson probability formula to calculate the probability of each event count.
- For cumulative questions, it adds the probabilities of all included counts.
- It highlights the included bars on the probability chart.
- It reports the mean, variance, standard deviation, and most likely event count.
- It explains the result in plain language so students understand what the probability means.
Formula & Equations Used
Poisson probability: P(X = k) = e^-λ λ^k / k!
Expected value: E(X) = λ
Variance: Var(X) = λ
Standard deviation: σ = √λ
Cumulative probability: add all included P(X = k) values.
Example Problem & Step-by-Step Solution
Example 1 — Find P(X = 6)
- A help desk receives an average of λ = 4 calls per hour.
- We want the probability of exactly 6 calls in one hour.
- Use P(X = k) = e^-λ λ^k / k!.
- Substitute λ = 4 and k = 6.
- The calculator highlights the bar for X = 6 and reports the probability.
Example 2 — Find P(X ≤ 6)
- Use the same call center setup: λ = 4.
- Choose P(X ≤ x).
- Enter x = 6.
- The calculator adds P(X = 0) through P(X = 6).
- The graph highlights all bars from 0 through 6.
Example 3 — Find P(3 ≤ X ≤ 7)
- A store gets an average of λ = 5 arrivals every 10 minutes.
- Choose between-values mode.
- Enter lower value 3 and upper value 7.
- The calculator adds the probabilities for 3, 4, 5, 6, and 7 arrivals.
Frequently Asked Questions
Q: What is a Poisson distribution?
A Poisson distribution gives the probability of a certain number of events occurring in a fixed interval when the average rate is known.
Q: What does λ mean in a Poisson distribution?
The value λ is the average number of events expected in the chosen interval.
Q: What is the difference between exact and cumulative Poisson probability?
Exact probability finds one count, such as P(X = 6). Cumulative probability adds multiple counts, such as P(X ≤ 6) or P(X ≥ 6).
Q: When should I use a Poisson distribution?
Use it for independent event counts over a fixed interval when the average rate is approximately constant.
Q: Can Poisson approximate a binomial distribution?
Yes. Poisson can approximate a binomial distribution when the number of trials is large, the success probability is small, and λ = np.
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