Poisson Probability Distribution Calculator
The Poisson probability distribution calculator helps predict the likelihood of a specific number of events occurring in a fixed interval. It's used in various fields like telecommunications (call arrivals), traffic analysis (accident prediction), inventory management (demand forecasting), and biology (mutation rates). This calculator simplifies complex statistical computations, making it accessible for students, researchers, and professionals to analyze rare event probabilities efficiently.
Formula
P(k) = (λk * e-λ) / k!
Where:
λ = average rate of occurrence
k = number of occurrences
e = Euler's number (~2.71828)
How to Use
1. Enter λ (average rate) in first field
2. Enter desired k value (number of events)
3. Click Calculate
4. Results show exact probability and cumulative probability
5. Interpret results based on your scenario
6. Use Clear to reset inputs
FAQs
1. What's the difference between Poisson and Binomial distributions?
Poisson models rare events in continuous intervals, while Binomial deals with discrete trials. Use Poisson when probability is small and sample size large (λ = np ≤ 7).
2. Can I use decimal values for λ?
Yes, λ can be any positive real number. The calculator accepts decimal values for precise rate inputs.
3. What if k is not an integer?
k must be non-negative integer. The calculator automatically rounds decimal k values to nearest integer.
4. What's considered a "rare event" for Poisson?
Events where probability of occurrence is low (typically λ ≤ 7) and number of trials is large (n ≥ 100).
5. How accurate is the calculator?
Results are precise to 10 decimal places using JavaScript's Math library functions for exponent and factorial calculations.
6. Can I calculate cumulative probabilities?
Yes, results show both exact P(X=k) and cumulative P(X≤k) probabilities for comprehensive analysis.
7. What's the maximum λ value allowed?
Technically unlimited, but extremely large λ values (λ>1000) may cause calculation delays due to factorial computations.
8. How to handle time-based events?
Convert your time period to match λ's time unit. E.g., if λ=2 calls/hour and you want 3 calls in 2 hours, use λ=4.
9. Limitations of Poisson distribution?
Assumes events are independent and constant rate. Not suitable for dependent events or varying rates.
10. What does P(X=0) mean?
Probability of zero events occurring in the interval. Calculated as e-λ since 0! = 1 and λ⁰ = 1.