LaTeX
P(X=k)=\frac{e^{-\lambda}\lambda^k}{k!}Variables
- λ: expected count
- k: nonnegative integer count
How to use this formula
Models counts of independent events in a fixed interval.
Important notes
- Assumes a constant average rate and independent occurrences.
Quick example
For λ=2, P(X=0)=e^{-2}.
Applicability, worked calculation, and verification
Assumptions and domain checks
- Assumes a constant average rate and independent occurrences.
- For the Poisson Probability Mass Function, every denominator must be nonzero, and the numerator and denominator must remain correctly grouped.
- The event model, conditioning information, independence assumptions, and probability range must match the problem.
Worked example
For λ=2, P(X=0)=e^{-2}.
Common mistakes
- When copying Poisson Probability Mass Function, keep the complete numerator and denominator grouped; a missing brace or parenthesis changes the result.
- Do not assume events are independent or mutually exclusive unless the problem states or proves that condition.
Continue the workflow
Use Poisson Probability Mass Function in your own work
- Check the domainMatch the variables and assumptions to the problem before substituting values.
- Copy the exact notationPreserve grouping, signs, and exponents in
P(X=k)=\frac{e^{-\lambda}\lambda^k}{k!}. - Edit or convertOpen the expression in the LaTeX editor, then export it for your document or web page.
Review and verification
Last reviewed: 2026-07-23
Automated quality check: Kept noindex until the missing evidence is supplied.
Formula references
- Introductory Statistics 2eOpenStax, Rice University — Reviewed probability and statistics definitions, notation, and formulas.
Frequently asked questions
What is the Poisson Probability Mass Function used for?
Models counts of independent events in a fixed interval.
Can I copy this formula as LaTeX?
Yes. Copy P(X=k)=\frac{e^{-\lambda}\lambda^k}{k!} or open it in the LaTeX editor.
What should I check before using it?
Confirm that each variable, unit, domain restriction, and assumption matches the problem.