Poisson Distribution Probability Calculator

Calculator

Example Data Table

Formula Used

The Poisson distribution estimates the probability of a count in a fixed interval.

• Exact probability: P(X = k) = e^-λ λ^k / k!
• Mean: E(X) = λ
• Variance: Var(X) = λ
• Standard deviation: σ = √λ
• Cumulative probability: P(X ≤ k) = sum of P(X = x) from x = 0 to k
• Right tail: P(X > k) = 1 − P(X ≤ k)
• Range: P(a ≤ X ≤ b) = sum of exact probabilities from a to b

How To Use This Calculator

1. Select direct lambda or rate multiplied by interval.
2. Enter the expected count or rate details.
3. Enter the primary count k.
4. Enter the second count b for range questions.
5. Choose the probability mode.
6. Select decimal places for the result.
7. Press the calculate button.
8. Review the result above the form.
9. Use CSV or PDF export when needed.

Poisson Probability In Statistics

Poisson Distribution Probability Calculator

A Poisson model estimates the chance of a count during a fixed interval. It works when events happen independently. It also assumes the average rate stays steady. Common examples include calls per hour, defects per batch, arrivals per minute, clicks per page, or accidents per month.

Why This Calculator Helps

Manual Poisson work can become slow when cumulative tails are needed. This calculator handles exact probabilities, less than, at most, greater than, at least, between, and outside range questions. It also accepts a direct mean value or builds the mean from rate and exposure time. That makes it useful for service planning, quality control, reliability checks, and classroom practice.

Interpreting The Output

The main probability shows the chance of the selected event. The percentage format gives the same answer in a familiar form. The expected count equals lambda. The variance also equals lambda. The standard deviation is the square root of lambda. The mode is usually the largest integer not exceeding lambda. When lambda is an integer, two adjacent modes may exist.

Practical Statistical Notes

A Poisson distribution is best for counts, not measurements. It is not ideal when events strongly influence each other. It may also fail when the rate changes during the interval. In those cases, split the interval, use a non homogeneous model, or choose another distribution. For large lambda values, the normal approximation can be helpful for quick checks.

Using Results Carefully

Small probabilities do not mean impossible events. They mean events are uncommon under the stated assumptions. Large tail probabilities can reveal overload risk, frequent defects, or unusual demand. Always compare the result with real observations. If observed counts keep exceeding expected counts, review the rate estimate. Better data usually gives a better lambda.

Advanced Options

The result table displays nearby probabilities for quick comparison. This helps users see how probability mass moves around the mean. CSV export supports spreadsheets. PDF export supports reports and assignments. The calculator keeps formulas visible, so each answer remains easy to audit. For reporting, keep the chosen interval clear. A rate per hour must match hours. A rate per day must match days. Mismatched units create misleading probabilities, even when arithmetic is correct in practice.

FAQs

1. What is lambda in a Poisson distribution?

Lambda is the expected number of events in the chosen interval. It is also the mean and variance of the distribution.

2. When should I use a Poisson model?

Use it for independent count events over a fixed time, space, area, or volume. The average rate should remain reasonably stable.

3. What does P(X = k) mean?

It means the probability that the count is exactly k. For example, it can mean exactly four calls in one hour.

4. What is cumulative Poisson probability?

Cumulative probability adds exact probabilities up to a count. P(X ≤ k) gives the chance of k or fewer events.

5. How do I calculate greater than probability?

The calculator uses the complement rule. It finds P(X > k) by calculating 1 − P(X ≤ k).

6. Can lambda be zero?

Yes. If lambda is zero, the model expects no events. Then P(X = 0) is one, and other exact counts are zero.

7. Is Poisson useful for quality control?

Yes. It can estimate defect counts when defects occur independently and the average defect rate is stable across similar units.

8. Why does the calculator show a nearby table?

The table helps compare neighboring counts. It shows where probability is concentrated and how cumulative probability changes across values.