poisson distribution calculator

What this poisson distribution calculator does

This calculator helps you find probabilities for counts of events that happen randomly over a fixed interval of time, space, distance, or area. It supports common Poisson use cases:

• Exact probability: probability of exactly k events.
• Cumulative probability: probability of at most or less than k events.
• Tail probability: probability of at least or greater than k events.
• Range probability: probability that event counts fall between two bounds.

Poisson distribution, explained simply

The Poisson distribution models how many times an event occurs in a fixed interval when events happen independently and at a roughly constant average rate.

Core formula

P(X = k) = (λ^k e^-λ) / k!

Where:

• λ = expected average count in the interval
• k = observed count (0, 1, 2, ...)
• e = Euler's number (~2.71828)

Key properties

• Mean = λ
• Variance = λ
• Standard deviation = √λ

How to use the calculator

• Enter your average rate λ.
• Select the probability type you want.
• Enter the event count k (and k2 for interval mode).
• Click Calculate to get the probability in decimal and percent form.

Real-world examples where Poisson fits

• Number of customer arrivals per minute
• Website errors per day
• Calls to a support desk per hour
• Machine failures per month
• Typos per page in a long manuscript

When Poisson is appropriate

Use Poisson when your data meets most of these assumptions:

• Events are counts (integers, never negatives).
• Events happen independently.
• The average rate remains stable over the interval.
• Two events cannot happen at exactly the same instant in the same tiny slice of interval (practically rare).

Common mistakes to avoid

• Using a non-integer value for k.
• Using a probability as λ instead of an average count.
• Ignoring changing rates (e.g., peak hours vs. off-hours).
• Applying Poisson to strongly overdispersed data (variance much larger than mean).

Quick FAQ

Can λ be zero?

Yes. If λ = 0, then P(X=0)=1 and all other counts have probability 0.

What if my variance is much larger than my mean?

Your data may not follow Poisson well. Consider alternatives like the negative binomial model.

Is this calculator exact?

For moderate values it computes exact Poisson values numerically. For very large λ in cumulative calculations, a normal approximation is used for stability and speed.

Bottom line

If you need fast, reliable probability estimates for event counts, this poisson distribution calculator gives a practical way to compute exact and cumulative outcomes without hand calculations.