Poisson Distribution Calculator
Calculate Poisson distribution probabilities for exact, cumulative, or tail values. The Poisson distribution models the number of events occurring in a fixed interval of time or space when events happen at a constant average rate. See also our Binomial Distribution Calculator, Normal Distribution Calculator, and Exponential Distribution Calculator.
How to Use the Poisson Distribution Calculator
The Poisson distribution is a discrete probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time, distance, area, or volume. It assumes events occur independently at a constant average rate λ (lambda). This distribution is particularly useful when counting rare events over a large number of opportunities.
To use this calculator, enter the average rate parameter λ (the expected number of events per interval) and the number of events k you want to find the probability for. Select whether you want the exact probability P(X=k), the cumulative probability P(X≤k), or the upper tail probability P(X≥k). The calculator will compute the result along with supporting statistics including the mean, variance, and standard deviation.
Common applications of the Poisson distribution include modeling the number of customers arriving at a store per hour, the number of defects per unit of manufactured product, the number of calls received by a call center per minute, the number of accidents at an intersection per year, and the number of mutations in a strand of DNA. The distribution is also used in queuing theory, reliability engineering, and insurance risk modeling.
Poisson Distribution Formulas
Probability Mass Function (PMF):
P(X = k) = (λ^k × e^(-λ)) / k!
Where:
λ = average rate (expected number of events)
k = number of events (non-negative integer)
e = Euler's number ≈ 2.71828
Mean and Variance:
Mean: μ = λ
Variance: σ² = λ
Std Dev: σ = √λ
Cumulative Distribution Function:
P(X ≤ k) = e^(-λ) × Σᵢ₌₀ᵏ (λⁱ / i!)
Mode:
Mode = ⌊λ⌋ (floor of λ) when λ is not an integer
Mode = λ - 1 and λ when λ is an integer
Example Calculation
A call center receives an average of 5 calls per minute. What is the probability of receiving exactly 3 calls in a given minute?
Given: λ = 5, k = 3
P(X = 3) = (5³ × e⁻⁵) / 3!
= (125 × 0.006738) / 6
= 0.842250 / 6
= 0.1404 (14.04%)
Mean: μ = λ = 5 calls per minute
Std Dev: σ = √5 = 2.236
Interpretation: There is approximately a 14% chance
of receiving exactly 3 calls in any given minute.
Poisson Probability Reference Table
| λ (Rate) | k (Events) | P(X = k) |
|---|---|---|
| 1 | 0 | 0.3679 |
| 1 | 1 | 0.3679 |
| 2 | 2 | 0.2707 |
| 3 | 3 | 0.2240 |
| 4 | 4 | 0.1954 |
| 5 | 5 | 0.1755 |
| 5 | 3 | 0.1404 |
| 10 | 10 | 0.1251 |
When to Use the Poisson Distribution
Use when:
- Counting the number of events in a fixed interval of time, area, or volume
- Events occur independently of each other
- The average rate of occurrence is constant
- Two events cannot occur at exactly the same instant
- Examples: calls per hour, accidents per month, typos per page, customers per day
Do NOT use when:
- The number of trials is fixed (use binomial instead)
- Events cluster or are contagious (one event makes another more likely)
- The rate changes over time (non-homogeneous process)
- You need to model time between events (use exponential distribution)
Additional Solved Examples
Example: Hospital Emergency Room
An ER receives an average of 4 trauma patients per night shift (8 hours). What is the probability of receiving exactly 6 patients tonight?
Lambda = 4, k = 6
P(X=6) = (e^(-4) x 4^6) / 6!
= (0.01832 x 4096) / 720
= 75.05 / 720 = 0.1042
Answer: There is approximately a 10.42% probability of receiving exactly 6 trauma patients. The ER should be staffed to handle above-average nights since P(X > 4) is substantial.
Example: Website Server Capacity
A web server receives an average of 3 requests per second. What is the probability of receiving more than 5 requests in a given second?
Lambda = 3
P(X > 5) = 1 - P(X ≤ 5)
P(X=0) = e^(-3) = 0.0498
P(X=1) = 3e^(-3) = 0.1494
P(X=2) = 4.5e^(-3) = 0.2240
P(X=3) = 4.5e^(-3) = 0.2240
P(X=4) = 3.375e^(-3) = 0.1680
P(X=5) = 2.025e^(-3) = 0.1008
P(X ≤ 5) = 0.9161
P(X > 5) = 1 - 0.9161 = 0.0839
Answer: There is about an 8.39% chance of receiving more than 5 requests in any given second. The server should be provisioned to handle burst traffic above the average.
Practice Questions
Question 1
A book has an average of 2 typos per page. What is the probability a randomly selected page has no typos?
Answer: P(X=0) = e^(-2) x 2^0 / 0! = e^(-2) = 0.1353. About 13.53% of pages are error-free.
Question 2
A call center receives 10 calls per hour. What is the probability of receiving 15 or more calls in a given hour?
Answer: Lambda = 10. P(X ≥ 15) = 1 - P(X ≤ 14). Using cumulative Poisson: P(X ≤ 14) = 0.9165. P(X ≥ 15) = 1 - 0.9165 = 0.0835 or about 8.35%.
Question 3
Traffic accidents at an intersection average 1.5 per month. What is the probability of at least one accident in a given month?
Answer: P(X ≥ 1) = 1 - P(X=0) = 1 - e^(-1.5) = 1 - 0.2231 = 0.7769. There is about a 77.69% chance of at least one accident per month.
Key Takeaways
- The Poisson distribution has a single parameter lambda (the average rate), and both its mean and variance equal lambda.
- The Poisson approximates the binomial when n is large and p is small (rule of thumb: n > 20, p < 0.05).
- To change the interval, scale lambda proportionally (e.g., 3 per hour = 6 per 2 hours).
- If Var(X) is much larger than Mean(X), the data may be overdispersed and the Poisson model may not fit well.
- The time between Poisson events follows an exponential distribution with rate parameter lambda.
Frequently Asked Questions
What are the conditions for a Poisson distribution?
The Poisson distribution requires: (1) Events occur independently of each other, (2) The average rate λ is constant over the interval, (3) Two events cannot occur at exactly the same instant, (4) The probability of an event in a small sub-interval is proportional to the length of the sub-interval. These conditions are often summarized as events occurring randomly and independently at a constant average rate.
What is the relationship between Poisson and binomial distributions?
The Poisson distribution is a limiting case of the binomial distribution when the number of trials n is large, the probability of success p is small, and the product λ = np remains moderate. Specifically, as n → ∞ and p → 0 with np = λ constant, Binomial(n, p) → Poisson(λ). The rule of thumb is to use Poisson when n ≥ 20 and p ≤ 0.05.
Why is the mean equal to the variance in a Poisson distribution?
This is a unique mathematical property of the Poisson distribution. Both the mean and variance equal λ. This property is useful for checking whether data follows a Poisson distribution: if the sample mean and sample variance are approximately equal, the data may be Poisson-distributed. When the variance exceeds the mean (overdispersion), a negative binomial distribution may be more appropriate.
How do I choose the right value of λ?
The parameter λ represents the average number of events per interval. Estimate it from historical data by dividing the total number of observed events by the number of intervals observed. For example, if a hospital emergency room sees 120 patients over 24 hours, λ = 120/24 = 5 patients per hour. Ensure the interval matches your question.
Can λ be a non-integer value?
Yes, λ can be any positive real number. While k (the number of events) must be a non-negative integer, the rate parameter λ can be 2.5, 0.7, or any positive value. For example, if a website averages 3.5 errors per day, you would use λ = 3.5 to model the number of errors on any given day.
When should I use Poisson vs exponential distribution?
The Poisson distribution models the count of events in a fixed interval (discrete), while the exponential distribution models the time between consecutive events (continuous). They are closely related: if events follow a Poisson process with rate λ, then the time between events follows an Exponential(λ) distribution. Use Poisson for "how many events?" and exponential for "how long until the next event?"