) From a recent statistical analysis for the last five
years, on an average there
are 4 (major) 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 ~
Poisson(l) approximately, where the
l» 4 accidents (monthly average
number of accidents). Find the probability that there will be 4 or
more air accidents
(1) in a given
month of the next year; Hint: The
required probability is P(X ³ 4).
(2) in 4 of the
first 6 months of the next year;
Hint: 1st 4th 5th 6th
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| | | 6C4 different ways to specify
p p p
p 1-p 1-p
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 = 4).
(3) for the first time in the next year in June;
Hint:Define Y = # of
months required to observe the first occurrence of the event A (i.e., 4 or more
air accidents occur in a month) (start from first month of next year). Notice
that if the first occurrence is in June, the event did not happen from January to May.
In other words, the event in this problem is (X<4, X<4, X<4, X<4,
(4) Compare the probabilities calculated
in (2) and (3), and why explain these probabilities are larger or smaller.