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.
- Mean and Variance — for a Poisson distribution, both the mean and variance are equal to the parameter λ. Example: If X ~ Po(λ), then E(X) = Var(X) = λ.
- Adding Poisson Distributions — the sum of two independent Poisson random variables is also a Poisson distribution. Example: If X ~ Po(λ) and Y ~ Po(μ), then X + Y ~ Po(λ + μ).
Exam Tips
Key Definitions to Remember
- Poisson Distribution: A model for the number of events in a fixed interval.
- Mean and Variance: Both equal to λ in a Poisson distribution.
Common Confusions
- Confusing the mean with the variance; they are equal in a Poisson distribution.
- Assuming events are not independent when they should be.
Typical Exam Questions
- What is the probability of exactly 3 events occurring in a Poisson distribution with λ = 2? Use the Poisson probability formula with r = 3 and λ = 2.
- How do you calculate the mean and variance of a Poisson distribution? Both are equal to λ.
- If X ~ Po(2) and Y ~ Po(3), what is X + Y? X + Y ~ Po(5).
What Examiners Usually Test
- Understanding of when to use a Poisson distribution.
- Ability to calculate probabilities using the Poisson formula.
- Knowledge of properties like mean and variance being equal.