🎓 Statistics & Data • Probability & Distributions
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Poisson Distribution Calculator
Calculates Poisson probability P(X = k) = (λ^k · e^(-λ)) / k! for event arrivals in fixed intervals.
Poisson Process Solver
Poisson Distribution: P(X = k)
Models event counts occurring at a constant average rate λ in fixed time intervals or regions.
Presets:
Exact Probability P(X = 4)
18.88%
Exact Decimal = 0.1888
P(X ≤ 4)72.54%
Mean Rate (λ)3.5
Std Dev σ = √λ1.8708
Poisson PMF Probability Distribution (λ = 3.5)
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 events arrive at a known constant average rate.
Formula & Step-by-Step Calculation
P(X = k) = (λ^k · e^(-λ)) / k!
λ = average arrival rate, k = number of observed events.
Worked Step-by-Step Examples
Example 1
Call center receives average 3.5 calls/minute (λ=3.5). Probability of exactly 4 calls?
Solution: P(X = 4) = 18.88%
• (3.5⁴ × e^(-3.5)) / 24 = (150.06 × 0.0302) / 24 = 0.1888
Common Real-World & Academic Use Cases
- ✓ Web server HTTP traffic request bursts
- ✓ Hospital emergency room hourly patient admissions
- ✓ Nuclear decay radiation particle counting
How to Use the Poisson Distribution Calculator
1
Input Rate Parameter λ
Enter average event frequency.
2
Input Event Count k
Enter target number of arrivals.
3
Read Probability
Inspect exact probability percentage.
Frequently Asked Questions
Q: What is unique about the mean and variance of a Poisson distribution?
In a Poisson distribution, the mean and the variance are both exactly equal to λ.