Suppose near miss incidents at a certain airport, occur according to a Poisson p

Question

Suppose near miss incidents at a certain airport, occur according to a Poisson process at a rate of lambda = 0.25 incidents per week and let X be the total number of incidents that occur in a 52-week period. State the probability distribution of X, including any unknown parameters. Find the probability that exactly 12 incidents occur in a 52-week period. Find the probability that at most 12 incidents occur in a 52-week period. Given that no incidents occur in the first 10 weeks, find the probability that at most 12 incidents occur in the following 42 weeks.

Explanation / Answer

Lembda = 0.25 incidents per week let X be the total number of incidents that occurs in 52 weeks

Here = 0.25 * 52 incidents = 13 incidents

(a) P(X) =(e-) (x) / x! = [(e-13 13x/ x!] ; X E R

   = 0, otherwise

(b) P(12) = (e-13) (1312) / 12! =0.1100

(c) P ( X<= 12) = it can be calculated by either adding for X = 0,1,2,3...12 or by poisson calculator

P (X<= 12) = 0.4631

(d) as there were no incidents in the first 10 weeks, so in next 42 weeks the probability of at max 12 incidents occuring, where on average 42 * 0.25 = 10.5 incidents can oocur.

P(X<= 12; 10.5) = 0.7419 (by poisson calculator)

so P(no incident in starting 10 week) = 0.4631/0.7419 = 0.6242

Here 0.4631 is the probability of at most 12 incidents occuring in 52 weeks.

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