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QUESTIONS: 2014; 2b, 3c 2013; 2aii, 2bi Poisson DISTRIBUTION λxe-λ P(X = x) = x! Used for discrete radom variables within a given time interval or spatial area for instance p(X > 3) For poisson distribution to be applicable the situation must meet the following conditions: The parameter of poisson distribution is: λ = mean or expected value Variables must be discrete (not continuous) Formulae for poisson distribution are given in the formula sheet The observation time period or spatial area must be fixed in advance The expected value is µ = λ The number of events occuring in each seperate interval must be independent The standard deviation is σ = √λ Events cannot occur simultaneously Remember the mean is proportional to the time period or spatial area in question The variance is V(X) = λ practice Question The mean number of correct answers a student can guess in a 1 minute online test with unlimited questions is 14.5. Each right answer occurs independently and is equally likely to occur at any second of the test. Calculate the probabilty the student will be able to guess at least 3 correct answers in the first 30 seconds of the test and justify your choice of distribution. Step One Step Two Choose the apropriate distribution model and justify your choice Identify the parameters Poisson distribution: The mean for 60 seconds, λ = 14.5 The mean for 30 seconds, λ = 7.25 There is a fixed time interval (30 seconds) Use binomial distribution tables in the formula sheet or a graphics calculator to find the probability Each event or correct question is Using tables or Ppd calculator function independent and events cannot occur p(X > 3) = 1 - (p(X=0) + p(X=1) + p(X=2)) Variables are discrete simultaneously = 1 - (0.000710 + 0.00515 + 0.0187) = 0.975 Using Pcd calculator function studytime.co.nz facebook.com/studytimenewzealand p(X > 3) = 1 - p(X < 2) = 1 - 0.0245 = 0.976