Statistical Methods

Unit 2 - Probability distributions

2.6.The Poisson Distribution
The Poisson distribution is another discrete probability distribution which has frequent applications faunal sampling operations where the character or variate under study is the number of animals or species per unit of observation. In practice, if the count data represent the number of rare events occurring within a given unit of time or space (or some volume of matter), the distribution of these counts can be described by the Poisson distribution. If ‘p’ the probability of occurrence an event is very small and ‘n’ the number of trails is very large such that np is constant, then Binomial distribution tends to follow Poisson distribution. A random variable is said to follow Poisson distribution if its mass probability function is given by,

Where e is the base of natural logarithm having a value of 2.7183, m is the mean of the distribution. If m is known this distribution can be completely determined and is called parameter of the distribution. An important characteristic of the Poisson distribution is that its variance is equal to the mean of the distribution. The Poisson distribution is positively skewed.

Fig (4): Poisson distribution

However, as m (= np, when n is large) increases, it will tend to normal distribution. In Poisson distribution, it is assumed that rare events occur randomly and independently. Some examples of Poisson variable are number of ships arriving in a harbour per hour, number of animals per square of plankton species, the number of machines breaking down daily in a fish processing plant. As variance equals mean in the case of Poisson distribution, the ratio of former to the latter (i.e. s²/m) can be used to determine whether the variable under study is randomly distributed or over-dispersed. Theoretically if this ratio is greater than 1, the population is over dispersed and Poisson distribution will not be suitable to describe this population.