Examples

Earthquake wait time

Suppose the San Andreas fault produces major earthquakes modeled by a Poisson process, with an average of 1 major earthquake every 100 years.

(a) What is the probability that there will not be a major earthquake in the next 20 years?

(b) What is the probability that three earthquakes will strike within the next 20 years?

Solution

(a)

Since the average wait time is 100 years, we set $\lambda =0.01$ earthquakes per year. Set $X\sim \mathrm{Exp}(0.01)$ and compute:

$$P[X>20]=e^{-\lambda \cdot 20}\gg \gg e^{-0.01\cdot 20}\gg \gg \approx 0.82$$

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(b)

The same Poisson process has the same $\lambda =0.01$ earthquakes per year. Set $X\sim \mathrm{Erlang}(\mathrm{3,0.01})$, so:

$$\begin{array}{c}{f}_X(t)=\frac{{\lambda }^{\ell }}{(\ell -1)!}{t}^{\ell -1}{e}^{-\lambda t}\\ \\ \gg \gg \frac{{(0.01)}^3}{(3-1)!}{t}^{3-1}{e}^{-0.01\cdot t}\gg \gg \frac{{10}^{-6}}{2}{t}^2{e}^{-0.01\cdot t}\end{array}$$

Now compute:

$$\begin{array}{c}P[X\le 20]={\int }_0^{20}{f}_X(x)dx\\ \\ \gg \gg {\int }_0^{20}\frac{{10}^{-6}}{2}{t}^2{e}^{-0.01\cdot t}dt\gg \gg \approx 0.00115\end{array}$$