Live model
Produce a count-by-count ledger without hiding the unlisted tail
Generate a review table for every outcome from zero through a chosen cap. Each row keeps point mass, cumulative mass, inclusive survival, and exact-outcome odds aligned, while the result cards disclose how much distribution remains beyond the table.
| Count | Exact PMF | Cumulative CDF | Inclusive survival | Odds against exact | Threshold flag |
|---|
Current calculation process
Formula, substitution, intermediate quantities, and check
p0=e^(-lambda); pk+1=pk lambda/(k+1); F(k)=sum(i=0...k)pi; S(k)=1-F(k-1)
The count rows retain their original Poisson probabilities and are never renormalized to the selected cap. Odds against each exact outcome are (1-p)/p; these are point-outcome odds and not threshold odds.
Use the table builder
Five steps for an auditable outcome ledger
- Define lambda and exposure. Record the business window alongside the number.
- Select a maximum that covers the decision. It must include the highlight threshold.
- Review omitted tail before export. Extend the cap if the remainder is material.
- Use the correct probability column. PMF is exact, CDF is at most, survival is at least.
- Save the ledger as evidence. Copy, TXT, and PDF all use the current rows rather than fixed examples.
Five foundations
Reading a discrete outcome table
1. Zero is a real outcome
The event-free row often carries substantial probability and starts the recurrence.
2. Columns answer different questions
Exact PMF, at-most CDF, and at-least survival must not be compared as if interchangeable.
3. Recurrence improves traceability
Each next mass equals the prior mass multiplied by lambda/(k+1), offering a row-level arithmetic check.
4. A finite list is not the support
The Poisson distribution extends to all non-negative integers even if the omitted tail becomes tiny.
5. A mode can tie
For positive integer lambda, counts lambda-1 and lambda have equal maximum mass; inspect the rows instead of assuming uniqueness.
Calculation anatomy
Columns, symbols, and default table construction
| Item | Definition | Audit use |
|---|---|---|
| pk | P(X=k) | one exact outcome |
| F(k) | sum through k | capacity through count k |
| S(k) | 1-F(k-1) | inclusive trigger at k |
| (1-pk)/pk | odds against exact k | point-frequency communication |
| M | maximum listed count | table truncation boundary |
| 1-F(M) | omitted upper tail | completeness check |
At lambda=6, row zero begins with e^-6. Row one multiplies by 6/1, row two by 6/2, and so on through count 15. The table total plus 1-F(15) must equal one.
Deep analysis
Three ways the ledger supports review
Capacity lookup
Find the smallest row whose CDF reaches a service target. That count is a discrete capacity quantile.
Escalation lookup
Read survival at the trigger row. Because it is inclusive, the threshold count is not accidentally omitted.
Truncation governance
Set an acceptable remainder before publishing a table. A 0.1% omitted tail may be trivial for teaching yet material for safety.
Decision cases
Documentation and truncation examples
Daily arrivals review
An analyst lists zero through fifteen arrivals at lambda=6, highlights ten and above, and exports exact, cumulative, and survival columns for a staffing meeting.
Maximum too close to the mean
If the table stops at six when lambda=6, a large upper tail remains. The visible omitted-mass card prevents the short ledger from being treated as a complete probability distribution.
Terms
Outcome-ledger vocabulary
- Outcome row
- One whole-number count and its probability measures.
- Recurrence
- Rule deriving p(k+1) from p(k).
- Listed mass
- Sum of PMF values through the selected maximum.
- Inclusive survival
- Probability of count k or greater.
- Odds against
- Ratio of all other outcomes to one exact outcome.
- Truncation
- Stopping a displayed infinite-support distribution at a finite row.
FAQ
Questions about Poisson outcome tables
Why start at zero?
Zero is in the Poisson support and anchors total mass.
What is inclusive survival?
P(X>=k), equal to one minus the CDF through k-1.
Why show exact-outcome odds?
They translate PMF into (1-p):p but do not describe a tail.
How high should the maximum be?
High enough that omitted mass is immaterial and all decision counts are included.
Is the truncated table normalized?
No. Original probabilities are retained and remainder is reported.
Can the mode tie?
Yes, when positive lambda is an integer, lambda-1 and lambda can share the maximum mass.
Limits and evidence
Table boundaries
- The table ends at a user-selected maximum while the mathematical support remains infinite.
- Odds apply to exact rows and are not odds of crossing the highlight threshold.
- Very small displayed probabilities are rounded, while calculations retain full precision.
- Poisson assumptions of fixed exposure, constant rate, and independent occurrences still apply.
- A table alone does not quantify uncertainty in an estimated lambda.
Sources and related tools