F09 - Further Special Discrete Probability Distributions - Lecture Notes (Student_s Version)

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Special Discrete Probability Distributions (Student’s version) Page 1 of 20 National Junior College 2016 – 2017 H2 Further Mathematics Topic F8: Further Special Discrete Probability Distributions Lecture Notes Key Questions to Answer: A. The Poisson Distribution 1. What is a Poisson random variable? 2. Why is a Poisson random variable discrete? 3. What are the conditions for a random variable to be modelled by a Poisson distribution? 4. What is the probability distribution function of this distribution? 5. How do we calculate the mean and variance of a Poisson distribution? 6. How do we calculate probabilities for the sum of two or more independent P oisson random variables? 7. How is the binomial distribution related to the Poisson distribution? B. The Geometric Distribution 1. What is a geometric random variable? 2. Why is a geometric random variable discrete? 3. What are the conditions for a random variable to be modelled by a geometric distribution? 4. What is the probability distribution function of this distribution? 5. How do we calculate the mean and variance of a geometric distribution? 6. What is the memoryless property of a geometric random variable? Previously, you have learnt a particular type of discrete probability distribution, called the bino mial distribution. In this chapter, we will explore two other discrete probability distributions, called the Poisson and geometric distributions. www.KiasuExamPaper.com 823

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Special Discrete Probability Distributions (Student’s version) Page 2 of 20 A. THE POISSON DISTRIBUTION §1 Definition and Theory Suppose you often receive WhatsApp messages and you are interested to know how likely you are to receive a certain number of WhatsApp messages in a certain day. Assuming that on average, you receive 9 WhatsApp messages each day, how do you find the probability that you receive, say, 2 WhatsApp messages in a particular day? To start off, we can partition the day equally into 24 disjoint one-hour periods, as illustrated in the diagram below: 1st hr 2nd hr 3rd hr … … 24th hr Assuming that you receive WhatsApp messages at a constant average rate throughout the day, t he probability that you receive a WhatsApp message in any of the 24 one-hour periods is 9 0.375.24  Let us now make the assumption that in every one-hour period, you either receive no WhatsApp message or only one WhatsApp message. (Realistically, this is not a valid assumption to make – but let us leave this issue aside for now.) We can then model the distribution for the number of WhatsApp messages that you receive in this particular day by a binomial distribution approximately, as follows: B(24,0.375)X  app