EXAMPLE 21Recognizing when to use the Poisson distribution

For each of the following situations, state whether or not the random variable follows a Poisson probability distribution. If not, state why not.

  1. is the number of telephone poles along a particular one-mile stretch of highway.
  2. is the number of rabbits living in a particular five-acre plot of land scheduled for development.
  3. is the number of calls to a radio station phone line in the hour following the announcement of a two-free-ticket giveaway to a sold-out concert.
  4. is the number of customers who purchase gasoline at a particular filling station from 2 P.M. to 2:15 P.M.

Solution

  1. does not follow a Poisson distribution. Telephone poles are spaced equidistantly from each other. Knowing the distance between the first two poles gives us the distance to each of the succeeding poles. This violates the requirement that the occurrences be random.
  2. does not follow a Poisson distribution. Related rabbits live together in warrens or dens. Thus, if there is one rabbit living within the specified area, there is probably more than one. This violates the independence requirement.
  3. does not follow a Poisson distribution. Presumably, the radio station would be inundated with calls within the first few minutes of the announcement. Later, there would be fewer calls to the station. This violates the requirement that the occurrences be uniformly distributed throughout the interval.
  4. does follow a Poisson distribution. The customers are random, independent, and occur uniformly over the 15-minute period.

NOW YOU CAN DO

Exercises 5–8.