Question

USA Today reported that Parkfield, California, is dubbed the world’s earthquake capital because it sits on top of the notorious San Andreas fault. Since 1857, Parkfield has had a major earthquake on average of once every 22 years.

a) Explain why the Poisson distribution would be a good choice for r = the number of earthquakes in a given time interval.

b) Compute the probability of at least one major earthquake in the next 22 years. Round lambda to the nearest hundredth, and use a calculator.

c) Compute the probability that there will be no major earthquake in the next 22 years. Round lambda to the nearest hundredth, and use a calculator.

d) Compute the probability of at least one major earthquake in the next 50 years. Round lambda to the nearest hundredth, and use a calculator.

e) Compute the probability that there will be no major earthquake in the next 50 years. Round lambda to the nearest hundredth, and use a calculator.

Answer 1

a) Consider the random variable

R : Number of major earthquakes in a given time interval.

i.e. we count the number of earthquakes in a given time interval.

r takes value 0,1,2,...........

Given that average number of major earthquake in 22 years is 1.

We know that for Poisson distribution Lambda is parameter it is also mean and variance of distribution.

Hence R ~ P ( )

The p.m.f. of R is

b) P ( at least one major earthquake in next 22 years) = P ( R > =1) = 1 - P (R = 0)

P ( at least one major earthquake in next 22 years) =0.6321

c) P ( There will be no major earthquake in next 22 years) = P ( R = 0)

P ( There will be no major earthquake in next 22 years) = 0.3679.

d) Since average number of major earthquake for 22 years = 1

Hence average number of major earthquake for 50 years = 50/22 = 2.2727

Hence R ~ P ( )

P ( at least one major earthquake in the next 50 years) = P ( R > =1 ) = 1 - P ( R= 0)

P ( at least one major earthquake in next 50 years) =0.8970

e) P ( There will be no major earthquake in next 50 years) = P ( R = 0)

P ( There will be no major earthquake in next 50 years) = 0.1030.