Calculate Poisson probabilities for a given rate (λ) and number of events (k): exact, cumulative, and upper-tail probabilities with mean and variance.
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0.2
P(X = 3)
0.43
P(X ≤ 3)
0.76
P(X ≥ 3)
| P(X = 3) — exact probability | 0.2 |
| P(X ≤ 3) — cumulative | 0.43 |
| P(X ≥ 3) — upper tail | 0.76 |
| P(X > 3) | 0.57 |
| Mean (λ) | 4 |
| Variance (λ) / SD (√λ) | 4 / 2 |
Also in Statistics
Probability Distributions
The Poisson distribution models the number of events occurring in a fixed interval of time or space, given a known average rate λ. Enter the average rate (λ) and the number of events (k) to calculate the exact probability, cumulative probability, and upper-tail probability.
The Poisson distribution is appropriate when events occur independently, at a constant average rate, and the probability of two events at exactly the same instant is negligible. Typical applications include: calls arriving at a call centre per hour, website visits per minute, bacteria colonies on a petri dish, defects per unit area.
The distribution is characterised by a single parameter λ (lambda), which equals both the mean and the variance.
The Poisson distribution is the limit of the binomial when n → ∞ and p → 0 with n·p = λ constant. In practice, use Poisson when counting events in a continuum (time, area, volume) with a very large or unbounded number of possible events.
Use binomial when there is a fixed number of trials n with clear success/failure outcomes.
Frequently asked questions
λ is the average rate of events in the interval of interest. For example, if an average of 4 customers arrive per hour, λ = 4. If you observe over a different interval, scale λ accordingly (e.g., λ = 8 for two hours).
Yes. λ is a rate and can be any positive real number. k (the number of events) must be a non-negative integer.
When λ is large (say λ > 30), the Poisson distribution is approximately normal with mean = λ and variance = λ. For small λ, use exact Poisson probabilities.
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