The Poisson Probability Distribution

1
The Poisson Probability Distribution

• The Poisson probability distribution provides a
  good model for the probability distribution of
  the number of rare events that occur randomly
  in time, distance, or space.

2
Poisson Probability Distribution

• Assume that an interval is divided into a very
  large number of subintervals so that the
  probability of the occurrence of an event in any
  subinterval is very small. Assumptions of a
  Poisson probability distribution
• The probability of an occurrence of an event is
  constant for all subintervals independent
  events
• You are counting the number times a particular
  event occurs in a unit and
• As the unit gets smaller, the probability that
  two or more events will occur in that unit
  approaches zero.

3
Poisson Probability Distribution

• The random variable X is said to follow the
  Poisson probability distribution if it has the
  probability function
• where
• P(x) the probability of x successes over a
  given period of time or space, given ?
• ? the expected number of successes per
  time or space unit ? gt 0
• e 2.71828 (the base for natural
  logarithms)
• The mean and variance of the Poisson probability
  distribution are

4
Example

• A life insurance company insures the lives of
  5,000 men of age 42. If actuarial studies show
  the probability of any 42-year-old man dying in a
  given year to be 0.001, the probability that the
  company will have to pay 4 claims in a given year
  can be approximated by the Poisson distribution.
• P ( X 4 \ n 5000, p 0.001 ) 0.1745