Poisson Distribution
P(X = k) = (λk · e−λ) / k!
Practical Examples
📞 Call Center
Average 3 calls/hour. What is P(exactly 2 calls)?
λ=3, k=2
🏭 Product Defects
Average 4.5 defects/batch. P(6 defects)?
λ=4.5, k=6
🛒 Customer Arrivals
Average 10 customers/hour. P(8 arrive)?
λ=10, k=8
🐛 Software Bugs
Average 2 bugs/1k lines. P(0 bugs)?
λ=2, k=0
⚡ Server Failures
Average 0.5 failures/month. P(1 failure)?
λ=0.5, k=1
📧 Email Volume
Average 20 emails/day. P(15 emails)?
λ=20, k=15
About 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 a known constant mean rate.
Conditions for Use:
• Events occur independently of each other
• The average rate λ is constant over the interval
• At most one event occurs in an infinitesimally small sub-interval
• We count occurrences, not non-occurrences
Key Properties:
• Mean = λ, Variance = λ (mean equals variance is a hallmark of Poisson)
• For large λ (typically > 10), Poisson approximates Normal N(λ, λ)
• Poisson is the limit of Binomial when n is large and p is small
The Poisson distribution is a discrete probability distribution that models the number of events occurring in a fixed interval of time or space, given a known constant mean rate.
Conditions for Use:
• Events occur independently of each other
• The average rate λ is constant over the interval
• At most one event occurs in an infinitesimally small sub-interval
• We count occurrences, not non-occurrences
Key Properties:
• Mean = λ, Variance = λ (mean equals variance is a hallmark of Poisson)
• For large λ (typically > 10), Poisson approximates Normal N(λ, λ)
• Poisson is the limit of Binomial when n is large and p is small