Poisson Probability Calculator evaluated example 1
Use the page’s labelled starter inputs to verify Poisson Probability Calculator.
Inputs- Expected event count (λ): 3
- Event count (k): 2
Evaluated result: P(X = 2) = 0.2240418077
Poisson Probability Calculator
Calculate point, cumulative and tail probabilities from an expected event count and a nonnegative integer count.
Poisson Probability Calculator
ResultP(X = 2) = 0.2240418077
Model and interpretation
λ is the expected count in a fixed interval and k is a nonnegative integer. The model assumes independent events at a constant rate; all counts must use the same interval. It does not justify that assumption for your data. 0 ≤ λ ≤ 10⁴; 0 ≤ k ≤ 10⁵; k ∈ ℤ.
- P(X = k)
- 0.224042
- P(X ≤ k)
- 0.42319
- P(X > k)
- 0.57681
- E[X]
- 3
- Var(X)
- 3
Method source: NIST/SEMATECH — Poisson DistributionReport a correction
What this calculator answers
Poisson Probability Calculator applies the registered method “P(X=k) = exp(−λ)λᵏ/k!; F(k) = P(X≤k); S(k) = P(X>k)” to the displayed Expected event count (λ), Event count (k) inputs. Both worked examples come from the production calculation engine, and NIST/SEMATECH — Poisson Distribution defines the cited method or convention; no hidden inputs or current external data are inferred.
Variables, defaults, and limits
- Expected event count (λ)
- Expected event count (λ) is a number input. The displayed starter value is 3; it is an example, not a hidden assumption.
- Default: 3
- Accepted values: Allowed range: 0 to 10,000. Numeric entries must be finite and unambiguous.
- Event count (k)
- Event count (k) is a number input. The displayed starter value is 2; it is an example, not a hidden assumption.
- Default: 2
- Accepted values: Allowed range: 0 to 100,000. Numeric entries must be finite and unambiguous.
Formula and calculation rule
R[poisson-calculator]=P(X=k) = exp(−λ)λᵏ/k!; F(k) = P(X≤k); S(k) = P(X>k); x₁=Expected event count (λ); x₂=Event count (k)Poisson Probability Calculator evaluates the registered expression “R[poisson-calculator]=P(X=k) = exp(−λ)λᵏ/k!; F(k) = P(X≤k); S(k) = P(X>k); x₁=Expected event count (λ); x₂=Event count (k)” with the validated inputs. Full calculation precision is retained until the result is formatted for display.
Worked examples
Poisson Probability Calculator evaluated example 2
Change one valid input or choice and evaluate Poisson Probability Calculator again to check that the result responds deterministically.
Inputs- Expected event count (λ): 2
- Event count (k): 0
Evaluated result: P(X = 0) = 0.1353352832
Assumptions
- Poisson Probability Calculator uses the visitor-entered values exactly as labelled; it does not retrieve private records or current rates.
- The calculation is limited to the declared relationship: R[poisson-calculator]=P(X=k) = exp(−λ)λᵏ/k!; F(k) = P(X≤k); S(k) = P(X>k); x₁=Expected event count (λ); x₂=Event count (k).
- Intermediate values are not rounded; display rounding is applied only at the presentation boundary.
- The deterministic calculation runs locally and does not upload calculator inputs to the application API.
- The result is an educational arithmetic result and excludes facts that are not represented by an input.
Validation and boundaries
- Every visible required input must be present; a missing value is never replaced with zero.
- NaN, positive or negative infinity, unsafe overflow, and a non-finite final result are rejected.
- Field-specific minimums, maximums, and choices apply to Expected event count (λ), Event count (k).
- Zero, negative values, and discrete counts are accepted only when the displayed field definition permits them.
Review and correction links
Common mistakes
- Confirm the meaning and unit of Expected event count (λ), Event count (k) before calculating; a numerically valid value can still use the wrong convention.
- Do not round intermediate values when checking the result, because early rounding can change the last displayed digits.
- Changing an unstated real-world assumption does not change the calculator until the corresponding displayed input is changed.
- Use the result as the answer to the displayed mathematical question, not to a different word problem with hidden conditions.
Frequently asked questions
- What question does Poisson Probability Calculator answer?
- Calculate point, cumulative and tail probabilities from an expected event count and a nonnegative integer count. The implemented relationship is R[poisson-calculator]=P(X=k) = exp(−λ)λᵏ/k!; F(k) = P(X≤k); S(k) = P(X>k); x₁=Expected event count (λ); x₂=Event count (k).
- Which inputs does Poisson Probability Calculator use?
- It uses the visible fields Expected event count (λ), Event count (k). No hidden value is substituted for an omitted required input.
- What happens when Poisson Probability Calculator receives an invalid or extreme value?
- The page rejects missing, ambiguous, out-of-range, or non-finite values and refuses to display a non-finite result.
- Why can Poisson Probability Calculator differ from another result?
- Different unit conventions, endpoint policies, fee or rate assumptions, formula domains, and premature rounding can produce a different answer.
- Can I use Poisson Probability Calculator as professional advice?
- No. It answers the displayed mathematical question and cannot account for omitted real-world facts.
- How can I verify the Poisson Probability Calculator result?
- Recalculate the two evaluated examples without rounding intermediate values and compare the declared formula or convention with NIST/SEMATECH — Poisson Distribution.
Source and review scope
NIST/SEMATECH — Poisson Distribution
Scope: NIST/SEMATECH — Poisson Distribution is used to check the formula, definition, or convention relevant to Poisson Probability Calculator. The citation does not supply current personal, lender, tax, medical, or market data.
Source checked: