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Unit 14: Poisson Probability Distribution




          14.1 Poisson Distribution                                                             Notes

          This distribution is used as a model to describe the probability distribution of a random variable
          defined over a unit of time, length or space. For example, the number of telephone calls received
          per hour at a telephone exchange, the number of accidents in a city per week, the number of
          defects per meter of cloth, the number of insurance claims per year, the number breakdowns of
          machines at a  factory per day, the number of arrivals of customers at  a shop per hour, the
          number of typing errors per page etc.




             Notes       Poisson Process

             Let us assume that on an  average 3 telephone calls are received  per 10 minutes at a
             telephone exchange desk and we want to find the probability of receiving a telephone call
             in the next 10 minutes. In an effort to  apply binomial  distribution, we can divide  the
             interval of 10 minutes into 10 intervals of 1 minute each so that the probability of receiving
             a  telephone call (i.e., a success) in each minute  (i.e.,  trial)  becomes  3/10  ( note  that
             p = m/n, where m denotes mean). Thus, there are 10 trials which are independent, each
             with probability of success = 3/10. However, the main difficulty with this formulation is
             that, strictly speaking, these trials are not Bernoulli trials. One essential requirement of
             such trials, that each trial must result into one of the two possible outcomes, is violated
             here. In the above example, a trial, i.e. an interval of one minute, may result into 0, 1, 2, ......
             successes depending upon whether the exchange desk receives none, one, two, ...... telephone
             calls respectively.

             One possible way out is to divide the time interval of 10 minutes into a large number of
             small intervals  so that the probability  of receiving two or  more telephone  calls in  an
             interval becomes almost zero. This is illustrated by the following table which shows that
             the probabilities of receiving two calls decreases sharply as the number of intervals are
             increased, keeping the average number of calls, 3 calls in 10 minutes in our example, as
             constant.
             Using symbols, we may note that as n increases then p automatically declines in such a
             way that the mean m (= np) is always equal to a constant. Such a process is termed as a
             Poisson Process. The chief characteristics of Poisson process can be summarised as given
             below:
             1.  The number of occurrences in an interval is independent of the number of occurrences
                 in another interval.
             2.  The expected number of occurrences in an interval is constant.
             3.  It is possible to identify a small interval so that the occurrence of more than one
                 event, in any interval of this size, becomes extremely unlikely.
          14.1.1 Probability Mass Function


          The probability mass function (p.m.f.) of Poisson distribution can be derived as a limit of p.m.f.
          of binomial distribution when          such that m (= np) remains constant. Thus, we can write

                                      r      n r                   r      n r
                                    m      m                ! n  m     m
                                n
                     P r     lim      C r  1         lim            1
                                 n  n      n        n    ! r n r  ! n  n




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