# ) From a recent statistical

*label*Statistics

*timer*Asked: Dec 8th, 2015

**Question description**

) 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 intensity

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).

**Ans.**

(2) in 4 of the first 6 months of the next year;

Hint: 1^{st} 4^{th} 5^{th} 6^{th}

| ×
| × |
× | ×
| | | _{6}**C**_{4} different ways to specify
4 months

p p p p 1-p 1-p

Event

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).

**Ans.**

(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, X<4, X≥4).

**Ans.**

(4) Compare the probabilities calculated in (2) and (3), and why explain these probabilities are larger or smaller.

**Ans.**