Poisson Distribution Calculator
PMF and tails for counts with rate lambda.
PMF and tails for counts with rate lambda.
- Grade level
- 10–12
- Curriculum band
- Difficulty
- Intermediate
- For this topic
- Time to master
- 2 weeks
- With daily practice
- Tools on page
- 10 on page
- Calculator + practice
Calculator
Parameters — Poisson Distribution Calculator
Result Summary
Results
Headline result
0.146525
Density or probability
0.238103
Left tail
0.908422
P(>x)
Closed forms where they exist; Simpson quadrature otherwise; display rounds to six decimals.
Statistics
Live readouts from your current parameters.
Headline
0.146525
density / probability
Visualizations
Density vs tails
Left-tail mass
Formula
PMF and tails for counts with rate lambda.
Alternative Form
Variables
| Symbol | Name | Unit | Description |
|---|---|---|---|
| f | Density / PMF | probability | Point probability or density |
| F | CDF | probability | Left-tail accumulation |
| params | Parameters | varies | Family-specific constants |
Explanation
Densities describe shape; CDFs accumulate area; tails complement to one.
Derivation
Discrete CDFs sum mass terms; continuous CDFs integrate densities — closed forms where they exist, Simpson quadrature otherwise.
Step-by-Step Calculation
Follow each step to understand how the result is calculated from your inputs.
Validate parameters against their legal ranges.
Evaluate the density or mass at the point.
Accumulate the left tail (sum or integrate).
Complement for the upper tail.
Round for display to six decimals.
State the finished answer.
Worked Examples
See how the calculator works with different scenarios.
Scenario
Run the page defaults once.
Inputs
Calculation
Evaluate density then tails
Result
See live result
Interpretation
Confirms the engine path end to end.
Understanding Your Result
Your Result: Headline result: 0.146525
Typical range: Densities integrate to 1; CDFs live in 0–1; tails complement.
What This Means
Density says how concentrated probability is near x; CDF says how much sits at or below x.
Why It Matters
P-values, control limits, reliability and risk all read tails of named distributions.
Professional Interpretation
Name the family with its parameters whenever you quote a tail area.
Recommendations
- •Plot the curve before quoting tails
- •Keep parameters in legal ranges
- •Report both density and CDF
Learn — The Complete Guide
Poisson Distribution Calculator evaluates one member of the standard distribution zoo at your point.
The core concept
Parameters fix shape; the point selects a slice; integration turns slices into probabilities.
- Densities shape; CDFs accumulate
- Parameters fix the family member
- Tails drive decisions
Derivations — why the formulas work
Why tails complement
- 1Total area equals 1
- 2Left plus right covers all
- 3Right = 1 - left
Where you meet this in real life
Learn this first — prerequisites in order
- 1Functions
- 2Probability basics
- 3Curve reading
Careers that use this daily
- •Analyst — risk tails
- •Engineer — reliability
- •Scientist — test stats
- •Actuary — loss models
Why it helps
- •Closed forms where possible
- •Numeric tails elsewhere
- •Live curve feedback
Best practices
- •Check support bounds
- •Quote family plus parameters
- •Use tails for decisions
Limitations — what this tool does not do
- •Single point per run
- •Numeric CDF carries small error
Curriculum map — where this topic lives in school
| Level | Grades | Standards | Skills |
|---|---|---|---|
| High school | 10–12 | S-ID, S-CP | Normal model · Tails |
| AP Statistics | 11–12 | AP | Named families · P-values |
Timeline — how this idea grew up
1713
Bernoulli trials
Binomial law published.
1809
Gaussian curve
Normal law for errors.
A little history
- • Poisson (1837) modeled rare counts.
- • Fisher named chi-square, t and F families.
Exam-day checklist
- • Name family + params
- • Draw and shade
- • Check support
Myths vs facts
Practical Tips
- •Check parameter bounds first
- •Move x to feel tails
- •Use CDF for at-or-below
- •Complement for at-or-above
- •Copy results with the share bar
Common Mistakes
Assumptions
This calculator makes the following assumptions to provide estimates:
Limitations
Be aware of these limitations when using this calculator:
Frequently Asked Questions
Answers to common questions about this calculator.
When Poisson?
Relation to binomial?
How do I use this calculator?
Does it show steps?
Can I print or share results?
Is my data stored?
Related Calculators
Explore more calculators related to this topic.
Related Guides
Deepen your understanding with these educational resources.
Glossary
Technical terms used in mortgage calculations explained in plain language.
- PDF/PMF
- Density or mass at a point.
- CDF
- Left-tail accumulation.
- Quantile
- Inverse CDF at a probability.
- Support
- Values with nonzero probability.
- Tail
- Probability beyond a point.
Authoritative References
Trusted sources and standards referenced by this calculator.
Applicable Standards
Diagrams & Illustrations
Visual intuition for every core idea. Select a card to enlarge it.
Did You Know?
Short intuition builders — great for class warm-ups.
Normal rules the zoo
Sums of anything become bell-shaped — the central limit theorem.
Poisson counts events
Calls, clicks and decays at constant rate.
Chi-square tests variance
Squared normals sum to chi-square.
Compare Side by Side
Three readings of one point — shape, below and above.
| View | Measure | Value | Note |
|---|---|---|---|
| Density | — | f(x) | shape |
| CDF | 0.238103 | left tail | P(<=x) |
| Tail | 0.908422 | right tail | P(>x) |
Recent Calculations
Your last runs on this page — stored only in this browser, never uploaded.
Poisson Distribution Calculator: 0.146525
Current inputs — live
Study Plan
A week-by-week path from prerequisites to mastery. Your checklist progress saves in this browser.
Prerequisites
- ▸Function notation
- ▸Basic probability
- ▸Reading curves
Mastery progress — 0%
- I quote family plus parameters
- I read tails, not just peaks
- I respect support bounds
- 1
Week 1 · ~15 min/day
Densities and tails
Goals
- Evaluate PDF/PMF
- Read CDF tails
- Match family to data
Practice
One family daily on this page.
Exam tips
- •State the family
- •Show z conversion
Practice Problems
0/3 solvedTry each problem first, peek at the hint only if stuck, then reveal the solution and mark it solved.
P(Z <= 1) for standard normal?
P(3 heads in 10 fair flips)?
Why do tails matter more than peaks?
Printable Worksheet
Generate a fresh problem set for class or homework. Print it, or copy the answer key separately.
- 1.18 × 47 =
- 2.8 ÷ 11 =
- 3.38 × 26 =
- 4.35 ÷ 4 =
- 5.21 − 16 =
- 6.2 + 26 =
- 7.14 ÷ 12 =
- 8.23 ÷ 3 =
- 9.49 ÷ 7 =
- 10.22 − 17 =
- 11.41 + 18 =
- 12.47 − 35 =
Formula Cheat Sheet
Every formula on this page in one printable table.
| Formula | Meaning | Example |
|---|---|---|
| P(X<=x) = CDF | Left tail | Phi(1) = 0.8413 |
| P(X>x) = 1 - CDF | Right tail | Complement |
| z = (x - mu)/sigma | Standardize | Any normal to N(0,1) |
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Page Information
2026-09-29
3.1.0-math
Free to use for educational and professional purposes.