Exam Questions - Poisson distribution
Exam Questions - Poisson distribution
Introduction to Poisson Distribution
- The Poisson distribution is a probability distribution applied when modelling the number of times an event appears within a fixed interval of time or space.
- It is named after the French mathematician Siméon Denis Poisson.
- Poisson distribution assumes that events occur independently and at a constant average rate within a given interval.
Identifying a Poisson Distribution
- If a question mentions rates, intervals, and independent events, it’s a strong clue towards a Poisson distribution scenario.
- Also, if the mean (λ) and the variance are equal then it is likely that the distribution is Poisson.
Calculation using Poisson Distribution
- The formula to calculate the probability of a certain amount (k) of events occurring within a fixed interval in a Poisson distribution is P(X=k) = [(λ^k) * e^(-λ)] / k!.
- In this formula, λ represents the average rate of occurrence per interval, e is Euler’s number (approximately 2.71828), and k! denotes the factorial of k.
Properties of Poisson Distribution
- The probability of two or more events happening in a very small interval is practically zero.
- The average rate (λ) is constant throughout the interval.
- The number of events in non-overlapping intervals are independent of each other.
Understanding and Interpreting Poisson Distribution
- In contrast to binomial distribution, there is theoretically no upper limit to the number of events that can occur in the given interval in a Poisson distribution.
- A Poisson distribution could be a handy model to consider when analysing the frequency of an event over time or space, such as the number of emails received in a day, count of cars passing through a junction in an hour etc.
- Real life data might not fit perfectly into a Poisson distribution due to factors such as non-constant rate, dependence of events etc., but can often still provide a helpful model.
Further Points
- Understanding what a Poisson distribution is and how to recognise one is the foundation for solving relevant problems.
- Mastery of Poisson calculations involves knowing and using the formula correctly, especially in the case of cumulative probabilities where multiple probabilities need to be summed.
- The ability to interpret a Poisin distribution in context can aid in correctly setting up and solving a problem, and also in understanding real-world applications of this distribution.