2. Aims: To be able to solve problems involving sums of independent Poisson random variables. To practice exam style questions Poisson distribution Lesson 3

3. If X and Y are independent Poisson random variables with parameters λ and Ф,Ф, respectively, then the random variable S, defined by S = X + Y is a Poisson random variable with parameter λ + Ф.Ф. Sum of Independent Poisson R.V’s Example 1 Friday night at a busy bar in Nottingham, the mean number of under age males trying to purchase a drink is 18 per hour. The mean number of under age females trying to do the same, is 6 per half hour. Assuming that these are observations from independent Poisson distributions, determine the probability that, on a randomly chosen hour one Friday evening, there are exactly 25 people under age trying to buy a drink. The mean number per hour, of under age people trying to purchase a drink is ___________ P(X = 25) =

4. Example 2 An nosey neighbour is looking out of his window. Both cars and lorries pass his window at random. On average there are 300 cars per hour, while the mean time between lorries is 6 minute. Determine the probability that exactly 30 vehicles pass the observers window in a five minute period. Mean number of lorries per hour is: Mean number of vehicles per hour is: Mean number of vehicles per 5 minutes is: So if X~ Po( ) P(X = 30) = Sum of Independent Poisson R.V’s

5. On average Cathy makes 2 spelling errors per page, whilst clever Emma only make one spelling error every 3 pages. Determine the probability that there are less 22 errors on 24 pages of writing, whereby Cathy has written 12 and Emma the other 12. On w/b

6. Do exercise 2F, qu 1,3,4 and 6 – 15 minutes. Last half hour group work on exam questions On average Oliver makes 1 careless error per question, whilst Ruth only make one careless error every 4 questions. Determine the probability that there are more than 14 errors on 20 question solutions, whereby Oliver has answered 10 and Ruth the other 10. On w/b