Live model
Inspect one count, both tails, and the mass that remains off-table
Enter the expected events for one fixed exposure and a whole-number decision count. The page separates an exact outcome from cumulative and inclusive-threshold questions so a service trigger is not confused with a single count.
| Count k | P(X = k) | P(X <= k) | P(X >= k) | Decision marker |
|---|
Current calculation process
Formula, substitution, intermediate quantities, and check
P(X = k) = e^(-lambda) lambda^k / k!; F(k) = sum(i=0...k) P(X=i); P(X >= k) = 1 - F(k-1)
The Poisson probability mass function assigns probability only to whole counts. The live table uses the same lambda as every result card, keeps unrounded values internally, and reports the probability above the final row instead of renormalizing a truncated list.
Use the calculator
Five steps from exposure definition to threshold decision
- Fix one exposure window. Define the hour, delivery route, server-day, or other unit over which lambda applies.
- Enter the expected count lambda. Use a rate already scaled to that same window; do not enter a probability.
- Choose the target count k. Use "exactly" for one outcome, "at most" for a capacity ceiling, and "at least" for an escalation trigger.
- Extend the ledger beyond the decision row. Raise the table maximum until the omitted tail is small enough for the review purpose.
- Export the current evidence. Save inputs, result cards, formula substitution, table, limits, and sources with the operational record.
Five foundations
What the Poisson count model says-and does not say
1. Exposure owns the rate
A lambda of 4.2 per shift is not interchangeable with 4.2 per hour. Rate and exposure must be scaled together before calculation.
2. Counts are discrete
The model permits 0, 1, 2, ... events. A fractional lambda is allowed because it is a mean; a fractional observed count is not.
3. Mean equals variance
For a Poisson variable, both equal lambda. Observed variance far above lambda signals clustering, hidden heterogeneity, or changing rates.
4. Tails are directional
"At least six" includes six and uses F(5) as the complement. "More than six" would instead subtract F(6).
5. The support never ends
Even when large counts are improbable, the mathematical support extends without a finite maximum. Omitted mass must remain visible.
Calculation anatomy
Symbols, units, and the default substitution
| Symbol | Meaning | Unit / domain |
|---|---|---|
| lambda | Expected events in one fixed exposure | events per chosen window, 0-100 |
| X | Random event count | non-negative integer |
| k | Decision count | whole events, 0-250 |
| F(k) | Cumulative probability through k | 0-1 |
| k! | Factorial count normalization | dimensionless |
Default point substitution: P(X=6)=e^-4.2 x 4.2^6 / 6!. The calculation retains full precision, then formats result cards as percentages. The table independently adds all listed PMF rows and reconciles that sum with its omitted upper tail.
Deep analysis
Three decisions hidden behind similar-looking probability labels
Exact staffing outcome
P(X=k) answers "how often exactly this many?" It is useful for one-bin workload planning but does not describe overload by itself.
Capacity compliance
P(X<=k) answers whether a capacity of k covers the period. Compare this cumulative service probability with a stated service target.
Escalation frequency
P(X>=k) measures how often the trigger itself or anything worse occurs. Using P(X>k) would understate an inclusive trigger.
Decision cases
Normal and boundary examples
Support desk escalation
A queue averages 4.2 priority incidents per night. The manager asks how often six or more occur, not merely how often exactly six occur. The upper-tail card is the staffing trigger; the exact card describes only one possible night.
Zero-rate commissioning period
For lambda=0, the only modeled outcome is zero: P(X=0)=1 and every positive count has zero probability. This is a mathematical boundary, not evidence that a real process can never fail.
Terms
Poisson distribution vocabulary
- Exposure
- The fixed opportunity window over which events are counted.
- Rate parameter lambda
- Expected events in that exposure; also the model variance.
- PMF
- Probability mass assigned to one whole-number outcome.
- CDF
- Total probability at or below a count.
- Inclusive survival
- Probability at or above a count, equal to 1-F(k-1).
- Omitted tail
- Probability beyond the last displayed table row.
FAQ
Questions specific to count-distribution work
When is a Poisson distribution appropriate?
When exposure is fixed, the rate is reasonably constant, occurrences are independent, and simultaneous events are negligible at the counting resolution.
Why are mean and variance both lambda?
That equality follows from the Poisson distribution. Material overdispersion or underdispersion in data is a diagnostic warning.
Is P(X>=k) the same as P(X>k)?
No. The first includes k; the second begins at k+1.
Why report omitted tail mass?
A finite table cannot list the infinite support. Reporting the remainder preserves total probability.
Can lambda be fractional?
Yes. It is an expected count. Only realized outcomes and k must be whole numbers.
What if events cluster?
Consider a negative-binomial or process-specific dependence model instead of assuming Poisson independence.
Limits and evidence
Assumptions to preserve with the result
- Constant event intensity within the defined exposure; no seasonality or trend is fitted here.
- Independent event occurrences; common-cause bursts can make the Poisson tail too small.
- Accurate and comparable exposure definitions across the data used to estimate lambda.
- Display rounding is not used in downstream sums; exported rows preserve more digits than the cards.
- The model estimates process probability, not uncertainty in an estimated lambda.
Sources and related tools