Poisson Distribution Calculator
Calculate probabilities for Poisson distribution with customizable parameters
Calculate Poisson Distribution
Average number of events per interval
Number of events to calculate probability for
Probability Results
Poisson Formula: P(X = x) = e-λ × λx / x!
With your values: P(X = 3) = e-5 × 53 / 3!
Distribution Properties
Probability Mass Function
Blue bars show probability for each number of occurrences. Selected value (x = 3) is highlighted in darker blue.
Common Applications
Customer Arrivals
Average 5 customers per hour
λ = 5, x = number of customers
Email Arrivals
Average 10 emails per day
λ = 10, x = number of emails
Website Hits
Average 50 hits per minute
λ = 50, x = number of hits
Defects
Average 2 defects per batch
λ = 2, x = number of defects
Key Properties
Mean and variance are both equal to λ
Standard deviation = √λ
Used for rare events with known average rate
Events must be independent
Rate must be constant over time
Understanding the Poisson Distribution
What is the Poisson Distribution?
The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given that these events occur at a known constant average rate and are independent of each other.
When to Use Poisson Distribution
- •Events occur independently
- •Average rate (λ) is known and constant
- •Events are rare but theoretically unlimited
- •Probability of an event in small interval is proportional to interval length
Formula and Parameters
P(X = x) = e-λ × λx / x!
- λ (lambda): Average rate of occurrence per interval
- x: Number of occurrences we want to find probability for
- e: Euler's number (≈ 2.71828)
- x!: Factorial of x
Note: Both λ and x must be non-negative, and x must be a whole number.
Real-World Examples
Call Center
A call center receives an average of 8 calls per hour. What's the probability of receiving exactly 10 calls in the next hour?
λ = 8, x = 10
Manufacturing Defects
A factory produces items with an average of 0.5 defects per item. What's the probability of finding exactly 2 defects in an item?
λ = 0.5, x = 2
Network Traffic
A server receives an average of 100 requests per minute. What's the probability of receiving exactly 95 requests in the next minute?
λ = 100, x = 95
Radioactive Decay
A radioactive sample emits an average of 3.2 particles per second. What's the probability of detecting exactly 4 particles in the next second?
λ = 3.2, x = 4