Poisson Distribution Calculator

Calculate probabilities for rare events in a fixed interval. Find PMF, CDF, and distribution parameters for count data.

Last updated: March 2026

Rate (λ - lambda)

Average events per interval

Events (k)

Number of occurrences

Results

P(X = 6)

0.104196

Probability of exactly 6 events

P(X ≤ 6)

0.889326

Cumulative probability

P(X > 6)

0.110674

Upper tail probability

Mean (λ)

4.0000

Variance (λ)

4.0000

Std Dev

2.0000

What is the Poisson Distribution?

The Poisson distribution models the probability of a given number of events occurring in a fixed interval of time or space, when these events occur independently and at a constant average rate. It's the go-to distribution for modeling rare events or count data.

The distribution is characterized by a single parameter λ (lambda), which represents both the mean and variance. This makes the Poisson distribution memoryless—the probability of future events doesn't depend on past occurrences. Examples include: number of emails per hour, radioactive decay events per second, customers arriving per minute, or typos per page.

A key property: for Poisson distributions, mean = variance = λ. If your data has variance much larger than mean, it's "overdispersed" and may not fit Poisson well. The distribution is discrete (k = 0, 1, 2, ...) and right-skewed for small λ, approaching normality as λ increases (>20).

How to Use This Calculator

Step-by-Step Guide

Determine λ (lambda): Calculate or identify the average rate of events per interval. Example: average 4 emails per hour → λ=4.

Choose k (events): Specify number of events you want the probability for. Must be non-negative integer (0, 1, 2, ...).

Calculate probabilities: PMF gives P(X=k), CDF gives P(X≤k), upper tail gives P(X>k). Distribution parameters shown below.

Interpret results: PMF tells exact probability, CDF tells cumulative probability up to k. Use for decision-making and risk assessment.

Key Formulas

PMF: P(X=k) = (λ^k × e^(-λ)) / k!

CDF: P(X≤k) = Σ[i=0 to k] P(X=i)

Mean: E[X] = λ

Variance: Var(X) = λ

Std Dev: σ = √λ

Example Calculation

Customer Service Calls

Scenario:

Call center receives average of 4 calls per hour
λ = 4 (rate parameter)
Question: What's the probability of exactly 6 calls in the next hour?

Calculate P(X = 6):
P(X=6) = (4^6 × e^(-4)) / 6!
P(X=6) = (4096 × 0.0183) / 720
P(X=6) = 75.01 / 720
P(X=6) ≈ 0.1042 or 10.42%

Calculate P(X ≤ 6):
Sum P(X=0) + P(X=1) + ... + P(X=6)
P(X≤6) ≈ 0.8893 or 88.93%

Calculate P(X > 6):
1 - P(X≤6) = 1 - 0.8893
P(X>6) ≈ 0.1107 or 11.07%

Interpretation:

There's a 10.42% chance of exactly 6 calls, 88.93% chance of 6 or fewer calls, and 11.07% chance of more than 6 calls in the next hour. This helps with staffing decisions and capacity planning.

Frequently Asked Questions

What is Poisson distribution?

Models count of rare events in fixed interval. Characterized by rate λ. Events occur independently at constant rate. Examples: emails/hour, customers/minute, defects/item. Discrete distribution for k=0,1,2,...

What is λ (lambda)?

Average rate of events per interval. Also equals both mean and variance (unique property). λ=4 means expect 4 events on average. Higher λ = more events expected. Must be λ>0.

When to use Poisson?

Use when: (1) counting events in fixed interval, (2) events independent, (3) constant average rate, (4) rare events (λ small relative to possible outcomes). Classic: radioactive decay, call center arrivals, typos per page.

PMF vs CDF?

PMF P(X=k): probability of exactly k events. CDF P(X≤k): probability of k or fewer events (cumulative). Upper tail P(X>k)=1-CDF: probability of more than k. Use CDF for 'at most' questions.

Mean = Variance property?

Unique to Poisson: E[X]=Var(X)=λ. If data variance ≫ mean → overdispersion, consider negative binomial. If variance ≪ mean → underdispersion, unusual for count data. Test: variance-to-mean ratio ≈ 1.

Poisson vs Binomial?

Poisson: unlimited trials, rare events, rate λ. Binomial: fixed n trials, probability p per trial. Poisson is limit of Binomial as n→∞, p→0, np=λ. Use Poisson when n large, p small.

What if λ is large (>20)?

Poisson approaches normal distribution for large λ. Can approximate P(X=k) using normal(μ=λ, σ²=λ). Makes calculations easier. Use continuity correction: P(X=k) ≈ P(k-0.5 < Y < k+0.5) for normal Y.

Real-world applications?

Quality control (defects per unit), epidemiology (disease cases per region), web analytics (clicks per minute), finance (rare events like defaults), queueing theory (arrivals), telecommunications (packet arrivals), ecology (species per quadrat).