Poisson Distribution On Calculator

Calculator

Lambda source

Lambda mean events

Rate per interval

Exposure intervals

Probability type

First event count x

Second event count y

Table start

Table end

Decimal places

Reset

Example Data Table

Scenario	Lambda	Question	Probability
Calls per hour	3	P(X = 5)	0.100819
Defects per batch	2	P(X <= 1)	0.406006
Errors per day	4	P(X >= 6)	0.214870
Arrivals per minute	1.5	P(2 <= X <= 4)	0.423599

Formula Used

The exact probability is:

P(X = k) = (e^(-lambda) * lambda^k) / k!

For the cumulative value, the calculator adds exact probabilities from zero through the selected count.

P(X <= x) = sum from k = 0 to x of P(X = k)

For a range, it adds all exact values from the lower count to the upper count.

P(a <= X <= b) = sum from k = a to b of P(X = k)

The mean is lambda. The variance is lambda. The standard deviation is the square root of lambda.

How to Use This Calculator

Select a direct lambda value, or choose rate and exposure.
Enter the average event count for the study interval.
Choose exact, cumulative, tail, or between probability.
Enter x. For a between question, also enter y.
Set the table range and decimal places.
Press Calculate. The result appears above the form.
Use CSV or PDF export for your saved report.

Understanding the Poisson Distribution

The Poisson distribution describes how often an event may happen in a fixed space, time, distance, area, or volume. It is useful when events are counted, independent, and driven by one average rate. The average rate is called lambda. It also represents the expected number of events.

This calculator supports exact, cumulative, tail, and interval probabilities. It can use a direct lambda value. It can also multiply a rate by an exposure interval. That helps when your source rate uses one time span, but your study uses another.

When to Use It

Use this tool when events are uncommon but countable. Typical cases include calls per hour, defects per batch, arrivals per minute, claims per month, goals per match, or errors per page. The model works best when the average rate is stable. It also assumes one event does not force another event to occur.

The result is a probability. It can be shown as a decimal or a percent. A small value means the selected count is unlikely under the chosen rate. A larger value means the count fits the assumed rate better. The mean and variance both equal lambda. The standard deviation is the square root of lambda.

Reading the Output

The exact option gives P(X = x). The at most option gives P(X ≤ x). The less than option gives P(X < x). The at least option gives P(X ≥ x). The greater than option gives P(X > x). The between option adds probabilities from the first event count to the second event count.

The table helps you inspect nearby event counts. It lists the probability at each count, the cumulative probability, and the upper tail. Export tools help you save results for reports, audits, class notes, or planning files.

Good Practice

Choose lambda carefully. It should match the same interval you want to study. If the average is per day, but the question is for three days, multiply the rate by three. Use whole numbers for event counts. Counts cannot be negative. Review the assumptions before making business, safety, or research decisions.

For sensitive uses, compare results with real records. Update the rate when demand, season, machine settings, or sampling rules change.

FAQs

What is lambda in a Poisson model?

Lambda is the average number of events expected in the chosen interval. It must match the same time, space, area, or volume used by the question.

Can lambda be a decimal?

Yes. Lambda can be a decimal because it is an average rate. Event counts must be whole numbers, but the average can be fractional.

What does P(X = x) mean?

It means the chance of seeing exactly x events in the selected interval, given the lambda value entered in the calculator.

What does cumulative probability mean?

Cumulative probability adds exact probabilities over several counts. For example, P(X <= 4) adds the chances for 0, 1, 2, 3, and 4 events.

When should I use the between option?

Use it when the question asks for a range, such as 2 to 6 arrivals. The calculator adds every exact count inside that range.

Why are mean and variance equal?

That is a key property of the Poisson distribution. When the model fits well, both the expected count and spread measure use lambda.

Can this calculator handle zero events?

Yes. Zero is valid as an event count. It is often important when checking the chance that no calls, defects, or arrivals happen.

Why use rate and exposure?

Use rate and exposure when the rate is given for one interval, but the question covers several intervals. The calculator multiplies them into lambda.