The preceding discussion explains why a Poisson random variable is usually a
good approximation for such diverse phenomena as the following:
- The number of earthquakes occurring during some fixed time span.
- The number of wars per year.
- The number of electrons emitted from a heated cathode during a fixed time
- The number of deaths in a given period of time of the policyholders of a life
- 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.
- (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
so the probability distribution function F of the random variable X is