A study in Kalb?eisch et al., 19914 reported that the number of warranty claims for one particular system on a particular automobile model within a year of purchase is well-modeled as a Poisson distributed random variable, X, with an average rate of λ = 0.75 claims per car.

(i) Determine the probability that there are two or fewer warranty claims on this speci?c system for a car selected at random.

(ii) Consider a company that uses the warranty claims on the various systems of the car within the ?rst year of purchase to rate cars for their initial quality. This company wishes to use the Kalb?eisch et al. results to set an upper limit, xu, on the number of warranty claims whereby a car is declared of "poor initial quality" if the number of claims equals or exceeds this number. Determine the value of xu such that, given λ = 0.75, the probability of purchasing a car which, by pure chance alone, will generate more than xu warranty claims, is 0.05 or less.