Normal Approximation To The Poisson Distribution
Let the random variable X denote the number of occurrences of an event in a particular interval of time and denote by A the expected, or mean, number of occurrences in that time interval. Then X obeys the Poisson distribution discussed in Section 4.7, with mean and variance
Consider now the situation in which the mean number of occurrences, A, is large. Suppose that the time interval of interest is broken down into subintervals of equal width, as in Figure 5.19. Then the total number of occurrences is the sum of the numbers of occurrences in each subinterval. Thus, we see that when the mean of the Poisson distribution is large, the total number of occurrences can be viewed as the sum of a moderately large number of random variables, each of which represents the number of occurrences in a subinterval of the time period. Hence, invoking the central limit theorem, we conclude that when A is large, the distribution of the random variable
is approximately standard normal.
As in the case of the binomial distribution, this result can be used to approximate probabilities. Here again, if A is of only moderate size, a continuity correction will be desirable.
FIGURE 5.19 Occurrences (•) in the interval from 0 to / broken down into subintervals of equal width
Approximating Poisson Probabilities Using the Normal Distribution
Let X be a Poisson random variable with mean A. If A is large, then to a good approximation
V VA VA
or, using the continuity correction
where Z has a standard normal distribution.
Post a comment