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IB DP Mathematics: Applications and Interpretation | SL & HL | Statistics and Probability | Topic 4.17
📋 Learning Intentions
- State the conditions under which a Poisson distribution is an appropriate model for a random variable.
- Define the Poisson distribution and identify its single parameter \( \lambda \) as both the mean and variance.
- Calculate exact and cumulative Poisson probabilities using the formula and the TI-84 Plus CE.
- Select the correct probability distribution (Poisson, binomial, or normal) given a real-world scenario.
- Apply the Poisson distribution to problems involving different time intervals or regions by rescaling \( \lambda \).
- Interpret Poisson probabilities in context, including inequalities such as \( P(X \geq k) \) and \( P(X \leq k) \).
Real-World Introduction
A hospital emergency department records an average of 4.2 patient arrivals per hour. No single arrival is predictable — each person arrives independently, and the rate stays roughly constant throughout the night shift. The question a hospital manager needs to answer is: what is the probability that more than 8 patients arrive in any given hour, requiring extra staff?
This kind of question — counting rare, independent events in a fixed interval — appears across science, engineering, and business. The number of radioactive decay events per second, the number of typos per page in a manuscript, the number of calls arriving at a call centre per minute: all share the same mathematical structure. The Poisson distribution is the model built precisely for this structure.
🌍 Why This Matters The Poisson distribution underpins queuing theory used by telecommunications engineers to design network capacity, by hospitals to schedule staffing, and by insurance companies to price policies for rare but costly events. Wherever a rate of occurrence is known but individual events are unpredictable, the Poisson model applies.
Core Concepts & Definitions
Before applying any probability distribution, the first task is always to check whether the conditions for that model are satisfied. The Poisson distribution is not universally applicable — it describes a very specific type of random process. Understanding the conditions is as important as knowing the formula.
📖 Key Definition — The Poisson Distribution A discrete random variable \( X \) follows a Poisson distribution with parameter \( \lambda > 0 \), written \( X \sim \text{Po}(\lambda) \), if it counts the number of events occurring in a fixed interval of time or space, subject to the Poisson conditions. The parameter \( \lambda \) is the mean (expected) number of events in that interval.
The Poisson distribution has a single parameter \( \lambda \) (the Greek letter lambda), which simultaneously equals the mean and the variance of the distribution.