From a recent statistical analysis for the last five years, on an average there

Question

From a recent statistical analysis for the last five years, on an average there are 5 air accidents per month in the world. Let X be the number of air accidents occurred in a randomly selected month. It is known that X approximately Poisson(lambda) approximately, where the intensity lambda 5 accidents (average number of accidents per month). Find the probability that there will be 5 or more air accidents in a given month of the next year; in 5 of the first 6 months of the next year; Define Y = # of months in the first 6 months of the next year in which "4 or more air accidents occur in a month". Y has a binomial distribution Compute P(Y = 5). (3) for the first time in the next year in April;

Explanation / Answer

In the given problem X follows poisson(5) distribution where X denotes the number of accidents in a random month.

(1) P(X>=5) = x>=5 e-5*5x/ x!

= x=04 e-5*5x/ x!

= e-5 [1+5+12.5+20.833+26.042] = 0.440 (approx)

(2) Let Y=Number of months of the 1st 6 months in the next year in which 5 or more accidents occur in a month.

Clearly Y~Binomial(6,p)

Here p = P(X>=5) = 0.440

Now P(Y=5) = 6C5*p5*(1-p)6-5 = 0.055 (approx)

(3) Let Z=The month in which more than 5 accidents ocuurs for the first time in the next year.

Clearly Z~Geometric(p)

Then the probability that 5 or more accidents occur in April for the first time in the next year is

P(Z=4) = (1-p)4-1*p = 0.077 (approx)

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