Spread of total yearly earthquake energy

Question

Earthquakes on a fault occur as a Poisson count $N$ averaging 8 per year. Each quake releases energy $X$ that is independent with mean 5 units and standard deviation 5 units, independent of the number of quakes. Let $S$ be the total energy released in a year.

What are the mean and variance of $S$ ?

Accepted Answer

This is a random sum (random number of quakes, each releasing random energy). The mean is expected count times expected energy: \mathbb{E}[S] = \mathbb{E}[N]\,\mathbb{E}[X] = 8 	imes 5 = 40 	ext{ units}. The variance combines both layers via the law of total variance: \operatorname{Var}(S) = \mathbb{E}[N]\operatorname{Var}(X) + \operatorname{Var}(N)\,\mathbb{E}[X]^2. For a Poisson count, \operatorname{Var}(N) = 8. With \operatorname{Var}(X) = 5^2 = 25 and \mathbb{E}[X]^2 = 5^2 = 25: \operatorname{Var}(S) = 8(25) + 8(25) = 200 + 200 = 400. Standard deviation \sqrt{400} = 20 units. Here the t…

Spread of total yearly earthquake energy

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