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I have the following data for earthquake magnitudes in a specific country in the years of 2006 to 2015 (years not mentioned had 0 earthquakes of magnitude equal or more than 7)

year         count
<int>        <int>
2011             1          
2013             2          
2015             1

the question I want to answer is the probability of an earthquake, happening with magnitude of equal or greater than 7 in the next 5 years.

I found this link about Gutenberg-Richter law but I can't find a good model to compute the probability.

I'm programming the probability in R and any help would be appreciated.

Alaleh
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1 Answers1

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Assuming Poisson distribution, a common choice of distribution for count data, the four earthquakes in ten years lead to maximum likelihood estimate of rate of earthquakes per year

$$\hat \lambda = \frac{\sum_{i=1}^n k_i}{n} = \frac{4}{10}$$

where $k_i$ is the number of earthquakes in the $i$-th year. This leads to estimating probability of any earthquake in a single year as

$$\Pr(K \ge 1) = 1 - \Pr(K=0) = 1 - e^{-\lambda}\frac{\lambda^k}{k!} = 1 - e^{-0.4} = 0.33$$

Assuming independence between years, the probability of any earthquake in five years is equal to

$$ \underbrace{1-\underbrace{\big(\Pr(K=0)\big)^5}_\text{no earthquake in 5 years}}_\text{opposite of the above} = 1-0.67^5=0.86 $$

Notice that this result is consistent with much simpler approach: observe that, according to your data, probability of observing a year with any earthquake is $3/10$, so probability of observing at least one such year in five is $1-(1-3/10)^5$.

I guess, there may exist more seismologically-correct approaches to such problem, but this is a simple probabilistic approach, since you asked on the stats site.

Tim
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