**Example**

The preceding discussion explains why a Poisson random variable is usually a good approximation for such diverse phenomena as the following:

**1.**- The number of earthquakes occurring during some fixed time span.
**2.**- The number of wars per year.
**3.**- The number of electrons emitted from a heated cathode during a fixed time period.
**4.**- The number of deaths in a given period of time of the policyholders of a life insurance company.

**Example**- Suppose that earthquakes occur in the western portion of the United States in
accordance with assumptions 1,2, and 3 with
and with 1 week as the
unit of time. (That is, earthquakes occur in accordance with the three
assumptions at a rate of 2 per week.)

**(a)**Find the probability that at least 3 earthquakes occur during the next 2 weeks.

**(b)**Find the probability distribution of the time, starting from now, until the next earthquake. *Solution:***(a)**We have**(b)**Let*X*denote the amount of time (in weeks) until the next earthquake. Because*X*will be greater than*t*if and only if no events occur within the next*t*units of time, we have*F*of the random variable*X*is given by