Study Notes
The Poisson distribution is used to model the number of times an event occurs in a fixed interval of time or space, given that these events happen with a known constant mean rate and independently of the time since the last event.
- Poisson Distribution — a probability distribution that expresses the probability of a given number of events occurring in a fixed interval of time or space. Example: The number of bacteria in a milliliter of solution occurring at a mean rate of 3 per milliliter.
- Parameter λ (lambda) — the average number of occurrences in the interval. Example: If a typist makes errors at a mean rate of 2 per page, λ = 2.
- Approximation to Binomial Distribution — used when the number of trials is large and the probability of success is small. Example: For a batch of 300 components with a 1 in 50 chance of being faulty, use Poisson approximation.
- Normal Approximation — used when λ is large, typically greater than 15. Example: For a Poisson distribution with λ = 24, use normal approximation.
Exam Tips
Key Definitions to Remember
- Poisson distribution: A model for the number of events in a fixed interval.
- Parameter λ: The mean number of occurrences in the interval.
Common Confusions
- Confusing the Poisson distribution with the binomial distribution.
- Forgetting to apply continuity correction when using normal approximation.
Typical Exam Questions
- What is the probability of no errors on a page if errors occur at a mean rate of 2 per page? Use Poisson formula with λ = 2.
- How do you approximate a binomial distribution with a Poisson distribution? Use Poisson when n is large and p is small.
- When can you use a normal distribution to approximate a Poisson distribution? When λ > 15.
What Examiners Usually Test
- Ability to calculate probabilities using the Poisson formula.
- Understanding when to use Poisson as an approximation to binomial.
- Applying normal approximation to Poisson distribution correctly.