If a city has an average of two accidents per day, how many accident-free days do you expect in a year?


Are we allowed to assume that the data come from a certain distribution? If yes, then this question can be answered. If no, then I can't think of a way to answer this question.

This data seems to come from a Poisson process, with a rate of lambda = 2. That is, accidents occur at a constant rate, accidents are independent and that there cannot be simultaneous accidents. The second and third assumptions are more questionable than the first, but I will just go with the Poisson assumption for now.

If lambda = 2, the probability that there are zero accidents in a given day is .135. The R code for this is $dpois(x=0,lambda=2)$. If we assume that there are 365 days in a year, the number of days without an accident is binomial with n = 365 and p = .135. Therefore, the expected number of days without an accident is 49.38.

The key step in this argument is to find the probability that there are zero accidents in one day.

  • 2
    $\begingroup$ +1 The distribution is not likely to be Poisson, though: we would expect sharp differences in accident rates (per unit time) between weekdays, weekends, and holidays, as well as seasonal variation and effects of temporal correlation. E.g., if the average is attained by virtue of a few hundred-car pileups each year, we might expect 300+ accident-free days annually. What can be said is that one very general model supposes the accident rate is a Poisson mixture; Jensen's Inequality then implies $365\exp(-2)\approx 49.397$ is a lower bound for the answer. $\endgroup$ – whuber Jun 24 '12 at 16:32

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