Poisson Distribution Calculator
Calculate Poisson probabilities for events occurring at average rate λ. Find P(X = k), P(X <= k), P(X >= k), mean, and standard deviation.
What is Poisson distribution?
Poisson distribution models the number of events occurring in a fixed interval of time or space, given a known average rate (λ). Requirements: 1) Events are independent. 2) Average rate is constant. 3) Two events cannot occur simultaneously. Examples: Customer arrivals (5 per hour), defects per product (0.2 per unit), emails received (10 per day), accidents per month. Formula: P(X = k) = (λ^k * e^-λ) / k!. Mean = λ, Variance = λ. Used in: Queuing theory, quality control, telecommunications, traffic flow.
How do I calculate Poisson probability?
Formula: P(X = k) = (λ^k * e^-λ) / k!. Where λ = average rate, k = number of events, e ≈ 2.71828. Example: Call center receives average 4 calls/hour. Find P(exactly 6 calls in next hour). λ=4, k=6. P(X=6) = (4^6 * e^-4) / 6! = (4096 * 0.0183) / 720 = 0.104 (10.4%). Cumulative: P(X <= k) = sum from i=0 to k. Real use: "At most 3 defects" = P(X <= 3), "More than 5 arrivals" = P(X > 5) = 1 - P(X <= 5).
When should I use Poisson vs Binomial distribution?
Use Poisson: Rare events, no fixed number of trials, events per interval, unknown total opportunities. Examples: Defects per unit, arrivals per hour, accidents per year. Use Binomial: Fixed trials (n), binary outcomes, known probability (p). Example: 100 coin flips. Poisson approximates binomial when n is large (>20), p is small (<0.05), and np < 10. Then λ = np. Example: 1000 items, 0.2% defect rate → Binomial complex, use Poisson with λ = 1000 * 0.002 = 2.
What does lambda (λ) represent in Poisson distribution?
Lambda (λ) = average rate of events per interval. It is both the mean and variance. Units match your interval: λ=5 calls/hour, λ=2.3 defects/product, λ=0.1 accidents/day. Calculating λ: From data: λ = total events / total intervals. From rate: λ = rate * time. Example: 20 customers in 4 hours → λ = 20/4 = 5 per hour. To find for 30 minutes: λ = 5 * 0.5 = 2.5. Higher λ = more frequent events. λ must be > 0. Distribution becomes more symmetric as λ increases.
How do I interpret Poisson distribution results?
P(X = k): Probability of exactly k events. P(X <= k): Probability of k or fewer (cumulative). P(X > k): Probability of more than k events. Example: Email λ=10/day. P(X=15) = 4.86% (exactly 15 emails unlikely). P(X <= 5) = 6.71% (quiet day unlikely). P(X > 15) = 8.39% (busy day moderately likely). Mean=Variance=λ=10, SD=sqrt10=3.16. Expect 10+/-3 emails typically (68% range: 7-13). Used for: Staffing (arrivals), inventory (demand), maintenance (failures), safety (incidents).