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  approximately. Then, the required probability can be approximated as   2 24 224P 2 (0.375) (1 0.375) 0.0012540995.2X        Now to address the issue that you may potentially receive more than one message in the same one - hour period, we can improve the model, we can partition the day into finer intervals, say into 1440 one-minute periods, as illustrated below: … In this case, the probability that you receive a message in each one -minute period is 9 1 ,1440 160 and therefore, 1B 1440, 160X       approximately. Therefore   2 1440 2 1440 1 1P 2 1 0.0049168056.2 160 160X                    Notice now that the approximate probability value has changed significantly. To yield better approximations, we can further partition the day into even finer intervals, say one-second intervals, which yield the following approximate distribution for X: www.KiasuExamPaper.com 824

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Special Discrete Probability Distributions (Student’s version) Page 3 of 20 1B 86400, 9600X       approximately. Therefore   2 86400 2 86400 1 1P 2 1 0.0049967376.2 9600 9600X                    Notice that the approximate probability value starts to change less significantly. If we continue this process and further partition the day into even finer intervals (of equal duration), we obtain the following approximate probability values: n p 9 n     approximate value for P(X = x) 100000 0.00009 0.004996923 10000000 0.0000009 0.004998085 600000000 0.000000015 0.004998097 Notice that the approximate values of P(X = 2) converges to a limit as ,n which is given by 2 2 2 2 2 2 2 9 9 ! 9 9 ( 9)lim 1 lim 1 12 2!( 2)! ! 9 9 ( 9)lim 12!( 2)! ( 1) 9 ( 9)lim 12! 9 n n n n n n n n n n n n n n n n n n n n n n n n n                                                                                        2 2 2 2 2 2 2 2 ( 1) 9 ( 9)lim 1( 9) 2! 9 ( 1) ( 9)lim lim 12! ( 9) 9 e 91 e =0.0049980971 2! 2! n n n n n n n n n n n n n n                                           [For the above calculations, we make use of the result that lim 1 e n x n x n          for any .]x  In general, one can prove the following result. Result 1.1 If  is a positive constant, then, elim (1 ) ,! x x n x n n p px x             where .p n  www.KiasuExamPaper.com 825

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Special Discrete Probability Distributions (Student’s version) Page 4 of 20 In general, random variables that count the number of times a particular event occur throughout a given region of time or space, such as the scenario illustrated above , i.e. the number of WhatsApp messages you receive in a day, can usually be modelled as having a probability distribution function whose rule can be expressed in the form as shown in the right-hand side of Result 1.1 , subject to certain conditions . Random variables having such a probability distribution function are called Poisson random variables, which is formally defined below. Definition 1.1 (Poisson Random Variable) A discrete random variable X which takes values 0, 1, 2, 3, ..., such that   eP ,! x X x x    (in MF26) where  is a positive constant, is said to be a Poisson random variable or to have a Poisson distribution with parameter . Notation: Po( ).X  Conditions for using a Poisson Model A Poisson random variable must satisfy the following conditions: No. Condition Description 1 Randomness Events occur randomly throughout the region of time or space. 2 Constant Mean Rate Events occur at a constant mean rate throughout the region of time or space, i.e. the average number of events occurring per unit time or space is a constant. 3 Independence Events occur independently of one another throughout the region of time or space. Examples of Poisson Random Variables  the number of particles emitted in a minute by a radioactive source,  the number of telephone calls within a fixed time interval at a switchboard,  the number of typographical errors in a randomly chosen page of a book,  the number of car accidents occurring on a particular stretch of road in a particular day. www.KiasuExamPaper.com 826

National Junior College Mathematics Department 2017 2016 – 2017 / H2 FMaths / Further Special Discrete Probability Distributions (Student’s version) Page 5 of 20 Result 1.2 If  is a positive constant, the function ef ( ) , 0,1, 2,3,! x x x x     is a probability distribution function. Proof: We need to show that (1) For all