Overview
This lecture introduces the Poisson distribution, its properties, conditions for use, and applications, with calculations and real-life examples.
Introduction to Poisson Distribution
- The Poisson distribution models the probability of a certain number of events occurring in a fixed interval, given a known average rate.
- It is a limiting case of the binomial distribution when the number of trials is large and probability of success is small.
- Discovered by Simon Poisson, originally applied to rare events like horse kick deaths in an army.
- Common in scenarios involving rare or independent events over time or space.
Conditions for Using Poisson Distribution
- Events occur in fixed intervals of time, space, or discrete units.
- The occurrence of each event is independent of others.
- The average rate (lambda, λ) remains constant across intervals.
- Events are rare with a low probability per interval.
- The variable of interest is a count within the interval.
Properties and Formula
- The shape of the Poisson distribution depends on λ (mean rate of events).
- For small λ (e.g., 1), the distribution is right-skewed; as λ increases (>5), it becomes more symmetric and bell-shaped.
- Formula: P(X = k) = e^(−λ) * λ^k / k!, where:
- X = number of events,
- k = specific value,
- λ = mean number of events,
- e = 2.718 (Euler's number).
- The mean (μ) and variance (σ²) of a Poisson distribution are both equal to λ.
- Standard deviation is the square root of λ.
Real-Life Applications
- Number of phone calls at a call center per time unit.
- Number of accidents at a location in a time period.
- Number of earthquakes in a region.
- Number of emails in a given time.
- Number of product defects in manufacturing.
- Number of customers arriving at a store, bank, or restaurant.
- Patients admitted to a hospital within a given period.
Example Calculations
- Given λ = 3, probability that only 1 person is killed by lightning in a year: P(1) ≈ 0.15 (15%).
- For 5% left-handed in 100 people, probability of zero left-handed: λ = 5, P(0) ≈ 0.0067 (0.67%).
- Probability of at most three left-handed: sum P(0), P(1), P(2), P(3) ≈ 0.265 (26.5%).
- Probability of more than three left-handed: 1 − sum P(0 to 3) ≈ 0.735 (73.5%).
Key Terms & Definitions
- Poisson Distribution — models the probability of a number of events occurring in a fixed interval when events happen at a constant average rate.
- Lambda (λ) — average rate of occurrence in the interval.
- Mean (μ) — expected number of occurrences, equal to λ.
- Variance (σ²) — measure of spread, also equal to λ.
- Standard Deviation — square root of λ.
Action Items / Next Steps
- Answer quiz questions on Poisson distribution properties and applications.
- Review and practice Poisson distribution example calculations.