Poisson Distribution Calculator
Enter an average rate (lambda) and a number of occurrences (x) to get the Poisson probability of exactly x events and the cumulative probability of x or fewer events.
What this calculator does
The Poisson Distribution Calculator estimates your result instantly from the values you enter, using standard statistics formulas. Enter an average rate (lambda) and a number of occurrences (x) to get the Poisson probability of exactly x events and the cumulative probability of x or fewer events. Calculations run in your browser, so nothing you type is sent to a server, and the output is an estimate rather than a certified or official figure.
Result
About Poisson Distribution Calculator
This calculator evaluates the Poisson distribution. Provide the average rate of events (λ) over an interval and a whole number of occurrences (x), and it returns the probability of exactly x events and the cumulative probability of x or fewer events.
Use it for counts of independent events in a fixed span of time or space, such as arrivals, defects or calls per period.
Formula & Mathematical Calculation
P(X = x) = f(Average rate (λ), Occurrences (x))
Calculated according to standard stats formulas implemented in the calculator core.
Calculation Example
Enter an Average rate (λ) of 3 and Occurrences (x) of 2, then click Calculate.
The tool returns P(X = x) = 0.224042 and P(X at most x) = 0.42319, with the note: “Exactly 2 events (λ = 3) has probability 22.4042%; cumulative P(X at most 2) = 42.319%.”
Key Benefits
- Returns both the exact point probability P(X = x) and the cumulative probability of at most x events
- Handles the special case of a zero rate, where all probability sits on zero events
- Uses a log-gamma factorial so it stays stable for larger counts
- Adds a note restating the probabilities as percentages for quick reading
How to Use this Calculator
-
Enter the average rate
Type the mean number of events per interval (lambda) into the Average rate field; it must be zero or positive. -
Enter the occurrences
Type the whole number of events you are asking about (x) into the Occurrences field. -
Calculate
Click Calculate to fill the P(X = x) box and the P(X at most x) box. -
Read the probabilities
Read the exact probability, the cumulative probability, and the percentage note beneath them.
Assumptions & Limitations
What the calculation assumes
- Every value you enter is treated as an exact, known amount.
- Any rate or interval you enter is assumed to hold for the whole period unless the form asks for it separately.
- Results use standard published formulas and account only for the values entered above.
- Rounding is applied for display; the underlying calculation uses your full precision.
- Results are returned in the same units you enter, unless the tool states the output unit.
What it does not cover
- Poisson Distribution Calculator produces a calculated result for reference and learning, not a certified or official measurement.
- Real-world outcomes differ when tolerances, rounding rules or local standards apply.
- Definitions and standards vary by region and change over time; verify the convention that applies to your case.
- Figures are only as accurate as the inputs you supply — check them before acting on the result.
- For work where the result carries real consequences, confirm it against an authoritative source.
Frequently Asked Questions
Free to use, no account needed. Calculations run in your browser, so nothing you type is sent to a server. Results are estimates — how we build and check our calculators.