Question

The number of recyclable cans carelessly tossed in a trash can during one day follows a Poisson distribution with λ=4. The Professor is a recycling devotee and at the end of each day she goes through all 50 trash cans near the hall to collect cans and put them where the belong in the recycling bin. Find the probability that she collects between 180 and 210 cans on a random night

Hint: You may find it helpful to review properties of the Poisson distribution and the Central Limit Theorem

Answer 1

Let x: no.of cans trash per can Given that lambda = 4 Let x ~ poisson (lambda = 4) Using property (additive) of poisson process (i) sigma xi ~ poisson (n lambda) i.e sigma_i = j^50 ~Poisson (n lambda = 200) (ii) using central limit theorem sigma xi ~ Normal (mu = n lambda, sigma = squareroot n lambda) i.e sigma xi ~ Normal (mu = 200, sigma = squareroot 200) Now p [180 < sigma xi < 210] = p [180 - mu/sigma < sigma xi - mu/sigma < 210 - mu/sigma] = p [180 - 200/squareroot 200 < z < 210 - 200/squareroot 200] = p [-1.4142 < z < 0.7071] = 0.6816 v