Question

3. The number of earthquakes per year in a certain region is a Poison random variable. The probability that no earthquake occurs in a given year in the region is 0.86. Find the probability that at least 2 carthquak es occur in a given year.

Answer 1

Let X be the random variable that represents the number of earthquakes in the region

Then

$\dpi{100} X ~ Poisson(\lambda)$

$\dpi{100} P(X=x) = \frac{e^{-\lambda} * \lambda ^ x}{x!}$

Now we are given P(X=0) = 0.86

Thus

$\dpi{100} P(X=0) = \frac{e^{-\lambda} * \lambda ^ 0}{0!} = 0.86$

$\dpi{100} \lambda = 0.1508228897$

We need to compute $\dpi{100} \Pr(X \ge 2)$ . Therefore, the following is obtained:

$\dpi{100} \Pr(X \ge 2) = 1 - \Pr(X < 2) = 1 - \Pr(X \le 1) )= 1 - 0.9897=0.0103$

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