During the years 1998–2012, a total of 29 earthquakes of magnitude greater than

Question

During the years 1998–2012, a total of 29 earthquakes of magnitude greater than 6.5 have occurred in Papua New Guinea. Assume that the time spent waiting between earthquakes is exponential.

a)What is the probability that the next earthquake occurs within the next three months?

b)Given that six months has passed without an earthquake in Papua New Guinea, what is the probability that the next three months will be free of earthquakes?

c)What is the probability of zero earthquakes occurring in 2014?

d)What is the probability that at least two earthquakes will occur in 2014?

Explanation / Answer

total months = 12 *14 = 168

rate = 29/168 = (1/ 5.79310) earthquake /month

a) P(X <= 3) = 1 - e^(- 1/ 5.79310*3)

= 0.4042042

b)

exponential distribution have memoryless property

hence

P(X > 3) = 1 - P(X< 3) = 0.59579

c)

there are 12 months in a year

hence

P(X > 12) = e^(- 1/ 5.79310*12) = 0.126

d)

number of earthquake is poisson distribution with mean = (1/ 5.79310) earthquake /month

hence for one year

rate = 29/14 = 2.0714285

hence

P(X>=2) = 1 - P(X = 0) - P(X = 1)

= 0.61298

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