# Poisson Distribution for Major Earthquakes

For a recent period of 100 years, there were 93 major earthquakes (at leat 6.0 on the Richter scale) in the world (based on data from the World Almanac and Book of facts). Assuming that the Poisson distribution is a suitable model, find the mean number of major earthquakes per year, then find the probability that the number of earthquakes in a randomly selected year is:

a=0

b=1

c=2

d=3

e=4

f=5

g=6

h=7

Here are the actual results:

47 years (0 major earthquakes)

31 years (1 major earthquakes)

13 years (2 major earthquakes)

5 years (3 major earthquakes)

2 years (4 major earthquakes)

0 years (5 major earthquakes)

1 year (6 major earthquakes)

1 year (7 major earthquakes)

Compare the actual results to those expected from the Poisson probabilities. Explain, how you do the comparison. Does the Poisson distribution serve as a good device for predicting the actual results?

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#### Solution Preview

Since there were 93 earthquakes in 100 years, we can estimate the average rate of earth quakes per year as 93/100 = 0.93

The poisson distribution is a discrete distribution where the probability is given by:

Probability(Number of events in 1 year = k) = rate^k x e^(-rate)/ k!

where e = 2.71828 i.e. base of the natural logarithm and ^ means raised to the power of and ! means factorial and

rate = 0.93 is the rate of events per year.

Let k denote the number of earthquakes in a year. In 1 year the:

Probability of 0 earth quakes (i.e. k = 0) is 0.93^0 x e^-0.93/0! = 0.39455

Probability of 1 earth quakes (i.e. k = 1) is 0.93^1 x e^-0.93/1! = 0.36693

Probability of 2 earth quakes (i.e. k = 2) is 0.93^2 x e^-0.93/2! = 0.17062

Probability of 3 earth quakes (i.e. k = 3) is 0.93^3 x e^-0.93/3! = 0.05289

Probability of 4 earth quakes (i.e. ...

#### Solution Summary

The solution uses a Poisson distribution to examine major earthquakes. The probability that the number of earthquakes in a randomly selected year is determined.