Question

By studying seismic measurements and geological evidence, scientists have made the following obser
vations about earthquakes. i. Small tremors (magnitude below 1.0 on the Richter scale) are almost constantly oc
curing on every continent.
ii. The number of earthquakes with magnitude at least 3.0 on the Richter scale averages
200,000 per year, worldwide.
iii. Based on geological evidence, the number of major earthquakes (magnitude> 7.0 on
the Richter scale) has averaged 20 per year, and this average rate has not changed over
the last 10,000 years.
iv. The number of earthquakes occurring each year is independent of the number that
occurred in any previous year.
v. The magnitude of an earthquake is inversely proportional to the logarithm of the
frequency that earthquakes of at least that magnitude occur. In other words, let
X(m) denote the expected number of earthquakes per year with magnitude greater
than or equal to m. Then logX(m) is proportional to m.
(a) Explain why it is reasonable to use a Poisson random variable to model the number of major
earthquakes occurring in any given period of time? Indicate which of the above observations
support your explaination.
(b) Let N(t) be a Poisson random variable that models the number of major earthquakes (magnitude
> 7.0) that will occur in the next t years. Give the probability mass function for N(t).
(c) Calculate the expected value and standard deviation for N(3).
(d) Calculate the probability that there will be at least 3 earthquakes in the next month.

Answer 1

1. It is reasonable to use poisson's random variable to model number of major earthquakes occuring per year because the number of major earthquakes per year is 20 , Here time interval is fixed,probability is very small and n is very large. Hence we use poisson random variable model here.Observation 3rd supports our answer.
2. Let N(t) be a poisson random variable which models the majot earthquakes(magnitude>7) occuring in next t years. Then probability mass function of this condition is

  

3. Expected value of N(3) =Mean = (Since all the moments of poisson distribution are equal to .

Variance Of N(3)= = 20

Standard Deviation of N(3) =