What is the Poisson distribution?

Models count of events in a fixed interval. Mean = λ = Variance. Used for rare events: calls, arrivals, defects.

Events occur independently at constant rate. Examples: call center arrivals, defects per unit, radioactive decay counts.

P(X=k) = e^(-λ) λ^k / k!. Probability of exactly k events.

When to use normal approximation?

When λ > 20, Poisson(λ) ≈ N(λ, √λ). Useful for continuity correction.

Binomial: n trials, fixed success prob. Poisson: count in interval, λ = expected count. Poisson approximates binomial when n large, p small.

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PROBABILITYDistributionsStatistics Calculator

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Poisson Distribution — Count of Events per Interval

P(X=k) for count data. Call centers, ER arrivals, software bugs. Mean = λ = Variance. Normal approximation when λ > 20.

Concept Fundamentals

λᵏe⁻λ/k!

Poisson PMF

Probability mass function

Mean = Variance = λ

Key Property

Single parameter

Rare event counting

Application

Arrivals, defects, calls

Siméon Poisson 1837

Developer

Discrete distribution

Compute ProbabilityPMF, CDF, between

Why This Statistical Analysis Matters

Why: Poisson models rare events in a fixed interval. Used in queueing, reliability, epidemiology. Mean equals variance.

How: Enter λ (mean rate). Choose mode: P(X=k), P(X≤k), P(X≥k), or P(a≤X≤b). Get probability, PMF, CDF charts.

●Mean = λ = Variance
●P(X=k) = e^(-λ)λ^k/k!
●λ>20 → normal approx

Sources:NIST/SEMATECH e-HandbookWolfram MathWorld - Poisson

STATISTICSDistributions

P(X=k) for Count Data — Events per Interval

PMF/CDF charts. Call centers, ER arrivals, software bugs, radioactive decay. Mean = λ = Variance. Normal approximation when λ > 20.

Exponential →Binomial →

Real-World Scenarios — Click to Load

Calculation Mode

Inputs

λ (mean rate)

k (count)

poisson.sh

CALCULATED

$ compute_poisson --lambda=5 --k=3 --mode=exact

Primary Probability

14.0374%

Mean (λ)

5.0000

Variance

5.0000

σ (√λ)

2.2361

Mode

5, 4

Normal approx

Poisson Distribution

P(X=3)

14.0374%

λ = 5Mean = 5.00σ = 2.24

numbervibe.com/calculators/statistics/poisson-distribution-calculator

PMF Bar Chart — P(X=k)

CDF Step Chart — P(X≤k)

Probability Table

k	P(X=k)	P(X≤k)	P(X≥k)
0	0.006738	0.006738	1.000000
1	0.033690	0.040428	0.993262
2	0.084224	0.124652	0.959572
3	0.140374	0.265026	0.875348
4	0.175467	0.440493	0.734974
5	0.175467	0.615961	0.559507
6	0.146223	0.762183	0.384039
7	0.104445	0.866628	0.237817
8	0.065278	0.931906	0.133372
9	0.036266	0.968172	0.068094
10	0.018133	0.986305	0.031828
11	0.008242	0.994547	0.013695
12	0.003434	0.997981	0.005453
13	0.001321	0.999302	0.002019
14	0.000472	0.999774	0.000698
15	0.000157	0.999931	0.000226
16	0.000049	0.999980	0.000069
17	0.000014	0.999995	0.000020
18	0.000004	0.999999	0.000005
19	0.000001	1.000000	0.000001
20	0.000000	1.000000	0.000000
21	0.000000	1.000000	0.000000
22	0.000000	1.000000	0.000000
23	0.000000	1.000000	0.000000
24	0.000000	1.000000	0.000000
25	0.000000	1.000000	0.000000
26	0.000000	1.000000	0.000000
27	0.000000	1.000000	0.000000
28	0.000000	1.000000	0.000000
29	0.000000	1.000000	0.000000

Showing first 30 rows.

Calculation Breakdown

INPUT

Parameters

λ=5, k=3

RESULT

Primary Probability

14.0374%

P(X = 3) = \frac{e^{-5} \cdot 5^{3}}{3!}

Mean

λ = 5.0000

E(X) = \text{lambda}

2.2361

√\text{lambda}

Normal approx

ext{Use} N(\text{lambda} ,√\text{lambda} ) ext{when} \text{lambda} >20

For educational and informational purposes only. Verify with a qualified professional.

📈 Statistical Insights

Mean = Variance = λ

— Definition

e^(-λ)

P(X=0) = e^(-λ)

— PMF

λ>20

Normal approximation valid

— NIST

Key Takeaways

The Poisson distribution models the count of events in a fixed interval when events occur independently at a constant average rate λ
PMF: P(X=k) = e^(-λ) × λ^k / k! — Mean = λ, Variance = λ, SD = √λ
Mode = ⌊λ⌋ (and λ-1 if λ is an integer)
When λ > 20, Poisson ≈ N(λ, √λ) — normal approximation is valid
Relationship to exponential: if events follow Poisson(λ), time between events is Exp(λ)

Did You Know?

📞Call center arrivals are modeled with Poisson — λ=5 per hour means on average 5 calls per hourSource: Queueing Theory

🌟Shooting star counts per hour follow Poisson distribution when the rate is constantSource: Astronomy

🐛Software bugs per release are often Poisson — λ=10 means ~10 bugs per release on averageSource: Software Engineering

🏥ER arrival rates are classic Poisson applications — used for staffing and capacity planningSource: Healthcare

☢️Radioactive decay: number of decays per second follows Poisson; time between decays is exponentialSource: Physics

📧Email counts per day can be Poisson when arrivals are independent and randomSource: Stochastic Processes

How It Works

1. The Poisson Process

Events occur randomly at a constant average rate λ per unit time/space. The number of events in a fixed interval is Poisson(λ).

2. The PMF Formula

P(X=k) = e^(-λ) × λ^k / k! — the probability of exactly k events. The factorial k! grows fast, so probabilities decay for large k.

3. Mean Equals Variance

E(X)=λ and Var(X)=λ — a unique property. If observed variance differs greatly from mean, Poisson may not fit.

4. Normal Approximation

When λ > 20, Poisson is well-approximated by N(λ, √λ). Use for quick hand calculations or when λ is large.

5. Exponential Connection

If events follow Poisson(λ), the time between consecutive events is Exp(λ). Poisson counts events; exponential models wait times.

Expert Tips

Poisson vs Binomial

When n is large and p is small, Binomial(n,p) ≈ Poisson(np). Use Poisson when n is unknown or very large.

Check Overdispersion

If variance >> mean, consider negative binomial. If variance << mean, data may be underdispersed.

λ Units Matter

λ must match your interval. λ=5 per hour means 5 events per hour; for 30 min use λ=2.5.

Queueing Theory

M/M/1 queues assume Poisson arrivals. Arrival rate λ and service rate μ determine queue length.

Why Use This Calculator vs Other Tools?

Feature	This Calculator	Excel	R	Manual
PMF + CDF charts	✅	⚠️	⚠️	❌
Normal approximation overlay	✅	❌	❌	❌
7 real-world presets	✅	❌	❌	❌
P(a≤X≤b) range	✅	⚠️	⚠️	⚠️
Step-by-step formulas	✅	❌	❌	✅
AI analysis	✅	❌	❌	❌

Frequently Asked Questions

When should I use the Poisson distribution?

When modeling counts of rare events in a fixed interval: call arrivals, defects, accidents, radioactive decays. Events must occur independently at a constant average rate λ.

What is the relationship between Poisson and exponential?

If events follow Poisson(λ), the time between consecutive events is Exp(λ). Poisson counts events; exponential models wait times.

When is the normal approximation valid?

When λ > 20. Use N(λ, √λ). For smaller λ, use exact Poisson formulas.

Why does mean equal variance for Poisson?

It's a mathematical property of the distribution. If your data has variance >> mean (overdispersion), consider negative binomial.

How do I estimate λ from data?

λ̂ = sample mean. The sample mean is the MLE for λ.

Poisson vs binomial — when to use which?

Binomial: fixed n trials, probability p. Poisson: count in interval, rate λ. When n large, p small, np=λ, they approximate each other.

Poisson Distribution by the Numbers

Mean = Variance

√λ

Standard Deviation

λ > 20

Normal Approx Rule

e^(-λ)

P(X=0)

Official Data Sources

NIST/SEMATECH e-Handbook ↗

US government statistical methods reference

Updated: 2025-12-01

Wolfram MathWorld - Poisson ↗

Mathematical properties and proofs

Updated: 2025-11-20

Wikipedia Poisson Distribution ↗

Comprehensive reference

Updated: 2026-01-15

Khan Academy ↗

Free statistics courses

Updated: 2026-01-15

SciPy poisson Documentation ↗

Python implementation reference

Updated: 2026-01-28

OpenIntro Statistics ↗

Free open-source statistics textbook

Updated: 2024-06-01

Disclaimer: This calculator provides Poisson probabilities for educational and professional reference. The Poisson model assumes events occur independently at constant rate λ. For critical applications (queueing, reliability, epidemiology), verify assumptions and consider overdispersion. When λ > 20, normal approximation is provided for comparison.

👈 START HERE

⬅️Jump in and explore the concept!

Poisson Distribution — Count of Events per Interval

Why This Statistical Analysis Matters

P(X=k) for Count Data — Events per Interval

Real-World Scenarios — Click to Load

Calculation Mode

Inputs

PMF Bar Chart — P(X=k)

CDF Step Chart — P(X≤k)

Probability Table

Calculation Breakdown

📈 Statistical Insights

Key Takeaways

Did You Know?

How It Works

1. The Poisson Process

2. The PMF Formula

3. Mean Equals Variance

4. Normal Approximation

5. Exponential Connection

Expert Tips

Poisson vs Binomial

Check Overdispersion

λ Units Matter

Queueing Theory

Why Use This Calculator vs Other Tools?

Frequently Asked Questions

When should I use the Poisson distribution?

What is the relationship between Poisson and exponential?

When is the normal approximation valid?

Why does mean equal variance for Poisson?

How do I estimate λ from data?

Poisson vs binomial — when to use which?

Poisson Distribution by the Numbers

Official Data Sources

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