Question

4. In a given city, 5% of all drivers get at least one parking ticket per year. Use the Poisson approximation to the binomial distribution to determine the probabilities that among 80 drivers (randomly chosen in this city): (a) 4 drivers will get at least one parking ticket in any given year; (b) anywhere from 3 to 6 drivers, inclusive, will get at least one parking ticket in any given year (c) at least 8 drivers will get at least one parking ticket within a two-year span; (d) Is a Poisson random variable a good approximation for this problem? Explain.

Answer 1

(d) The Poisson approximation to the Binomial distribution works well only when n is large and p is small. In general, the approximation works well if n ? 20 and p ? 0.05, or if n ? 100 and p ? 0.10. As a result, we can say that the Poisson random variable is a good approximation for this problem.

Let x: random variable denoting the no of drvers getting at least 1 parking ticket per year Given p = prob of getting at least 1 ticket = 0.05 n = 80 therefore X tilde Binomial (n = 80, p = 0.05) we do poission approximation, the lambda = np = 80 times 0.05 = 4 therefore X asymp poission (4) therefore The point of X is: p(x = x) = e^-4 4^x/x!, x = 0,1,2 P(x = 4) = e^-4 4^4/4! = 0.1954 P(3 lessthanorequalto x lessthanorequalto 6) = P(x = 3) + p(x = 4) + P(x = 5) + P(x = 6) = 0.6512 P(x greaterthanorequalto 8) = 1 - P(x < 8) = 1 - P(x lessthanorequalto 7) = 1- [P(x = 0) + P(x = 1) + + P(x = 7)] = 1- 0.9989 = 0.0511 The Poisson approximation to the Binomial distribution works well only when n is large and p is small.