Poisson Distribution and Median

Part (a) is definitely incorrect. The correct answer is

 Pr[A<3] = 0.1247.  

You can verify this in R - The Statistical Computing Platform:

 ppois(q=2, lambda=5)

where ppois() gives the cumulative probability, i.e. the case where k=0,1,2.

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Part (b) To elaborate on @deinst excellent comment: If the distribution is assumed to have a mean of 5.5 accidents per week, then $\lambda=5.5$, but median = 5. That is, while the mean, and therefore expected value, can be a non-intger, the median, as the middle value will always be an integer.

If, however, $\lambda=5.7$, then median = 6.

Interestingly, a closed form for the median of a Poisson distribution is not simple, so we have bounds and an approximation.

Again, in R, the median of a poisson distribution with rate r can be given by:

 mpois <- function(r) { floor(r + 1/3 - 0.02/r) }

so mpois(5.7) is seen to be 6.

The reference for this approximation can be found in Wikipedia: Poisson Properties, referring to a 1994 paper Choi KP (1994) On the medians of Gamma distributions and an equation of Ramanujan. Proc Amer Math Soc 121 (1) 245–251