Poisson Distribution Calculator

Calculate Poisson probabilities for rare events: P(X = k) = e^(-λ) λᵏ / k!. Find the probability of exactly k events occurring in a time interval given average rate λ.

📊 Statistics📐 P(X=k) = e⁻λ × λᵏ / k!🔢 Math
Average rate λ (expected events per interval)
Number of events k
Probability type
Please enter valid values.

Formula & Reference

VariableSymbolFormulaUnits
Poisson Distribution Calculator—P(X=k) = e⁻λ × λᵏ / k!probability

Step-by-Step Examples

Example 1
Call Center

Average 3 calls per minute (λ=3). P(exactly 2 calls in one minute)?

  • P(X=2) = e⁻³ × 3² / 2! = 0.0498 × 9 / 2
  • = 0.2240
  • 22.4% chance of exactly 2 calls in a minute
  • Mean and variance both = λ = 3
✓ P(X=2) = 22.4%
Example 2
Website Traffic

λ=10 visits per hour. P(15 or more visits in an hour)?

  • P(X≥15) = 1 - P(X≤14)
  • = 1 - Σ P(X=k) for k=0 to 14
  • ≈ 1 - 0.9165 = 0.0835
  • 8.35% chance of 15+ visits
✓ P(X≥15) = 8.35%
Example 3
Rare Events

Hospital: average 0.5 accidents per day (λ=0.5). P(no accidents)?

  • P(X=0) = e^(-0.5) × 0.5^0 / 0! = e^(-0.5) = 0.6065
  • 60.65% chance of no accidents on a given day
✓ P(X=0) = 60.65%

Real-World Applications

📞
Call Centers
Model call arrivals for staffing optimization. Queuing theory uses Poisson process.
🚗
Traffic Engineering
Vehicles arriving at an intersection modeled as Poisson process.
💻
Computer Networks
Data packets arriving at a router or server follow Poisson distribution.
🏥
Healthcare
Rare disease incidence in a population, equipment failures, ER arrivals.

Common Mistakes to Avoid

⚠️
Using Poisson when events are not rare

Poisson works best when p is small and n is large (λ = np). For frequent events with p near 0.5, use binomial.

⚠️
Forgetting λ is per interval

λ must match your time/space unit. If λ=6 per hour but you want per half-hour, use λ=3 for that calculation.

⚠️
Ignoring over-dispersion

Real data often has variance > mean (over-dispersion). Poisson assumes variance = mean. Negative binomial is better for over-dispersed count data.

Frequently Asked Questions

What is the Poisson distribution? ▾
Models the number of events occurring in a fixed interval (time, space) when events occur at a constant average rate λ and independently of each other. Mean = Variance = λ.
What is the Poisson process? ▾
A process where events occur independently at a constant average rate. Between any two events, the waiting time follows an exponential distribution with rate λ.
When does Poisson approximate binomial? ▾
When n is large (>100) and p is small (<0.01), so λ=np is moderate. P(X=k) ≈ e^(-λ)λ^k/k! is computationally simpler than the binomial formula.
What is the relationship between Poisson and exponential? ▾
The time between consecutive Poisson events follows an exponential distribution with rate λ. If events arrive at λ=3/hr, the waiting time between events is Exp(3): mean 1/3 hr = 20 min.
What is the Poisson regression? ▾
Regression model where the response variable is count data (assumed Poisson-distributed). Used for accident counts, disease cases, website visits. Part of the generalized linear model family.

Related Math Calculators

📊 Binomial Distribution📊 Normal Distribution📊 P-Value📊 Probability📊 Expected Value

Formula Explorer connections

Interpretation: This formula summarizes data, models uncertainty or supports inference about a population or random process. Assumption: The sampling design and distribution assumptions must match the data. Independence, sample size, outliers and measurement quality can materially affect interpretation.

Probability Calculator – Event Probability and Odds →Sample Size Calculator →Scatter Plot Calculator →Math Formula Explorer →