I should solve the following task:
In a certain city, earthquake occurrences are modeled by a Poisson-process with intensity $\lambda $ = 2 per year. The damage caused by such an earthquake, is assumed to be exponentially distributed with parameter $ \mu $ = 10³ Euros. What is the expected damage per decade?
- First, I calculated the expected number of earthquakes in a decade. Since Poisson-process, I used the following expectation: E(# eq's per decade) = $ \lambda $t = 2*10 = 20
- Then, I computed $ \lambda $ of the exponential distribution: $ \mu = \frac{1}{\lambda}$, therefore I get $\lambda = \frac{1}{10^3}$
- To get the expected damage, I calculated the expected value like this (x = damage):
$E(x) = \int_{0}^{20} x f(x) dx = \int_{0}^{20} x \lambda \exp(-\lambda x) dx$
I tried to solve this expectation by partial integration, but got in the end no meaningful solution.
So, I wonder, if my assumptions are incorrect or the partial integration failed? Can somebody help please?
Thanks!
Let $\{N(t):t\geqslant 0\}$ be the Poisson process governing earthquake occurrences and $\{U_n: n=1,2,\ldots\}$ the magnitude of each earthquake. Then $V_t:= \sum_{n=1}^{N_t}U_n$ is a compound Poisson process and by the tower property of conditional expectation, $$ \mathbb E[V_t] = \mathbb E[\mathbb E[V_t\mid N_t]] = \mathbb E[\mathbb E[U_1]\mathbb E[N_{10}] = \mathbb E[U_1]\mathbb E[N_{10}] = 1000\cdot2\cdot10 = 20000. $$