Historical Note:• Discovered by Mathematician Simeon Poisson in France in 1781.• The modelling distribution that takes his name was originally derived as an approximation to the binomial distribution.
Defination:• Is an eg of a probability model which is usually defined by the mean no. of occurrences in a time interval and simply denoted by λ.
Uses:• Occurrences are independent.• Occurrences are random.• The probability of an occurrence is constant over time.
Sum of two Poisson distributions:• If two independent random variables both have Poisson distributions with parameters λ and μ, then their sum also has a Poisson distribution and its parameter is λ + μ .
The Poisson distribution may be used to model a binomial distribution, B(n, p) provided that • n is large. • p is small. • np is not too large.
F o r m u l a:• The probability that there are r occurrences in a given interval is given byWhere, = Mean no. of occurrences in a time interval r =No. of trials.
The Poisson distribution is defined by a parameter, λ.
Mean and Variance of Poisson Distribution• If μ is the average number of successes occurring in a given time interval or region in the Poisson distribution, then the mean and the variance of the Poisson distribution are both equal to μ. i.e. E(X) = μ & V(X) = σ2 = μ
Examples:1. Number of telephone calls in a week.2. Number of people arriving at a checkout in a day.3. Number of industrial accidents per month in a manufacturing plant.
Graph :• Let’s continue to assume we have a continuous variable x and graph the Poisson Distribution, it will be a continuous curve, as follows: Fig: Poison distribution graph.
Example:Twenty sheets of aluminum alloy were examined for surface flaws. The frequency of the number of sheets with a given number of flaws per sheet was as follows: What is the probability of finding a sheet chosen at random which contains 3 or more surface flaws?
Generally,• X = number of events, distributed independently in time, occurring in a fixed time interval.• X is a Poisson variable with pdf:• where is the average.
Application:• The Poisson distribution arises in two ways:
1. As an approximation to the binomial when p is small and n is large:• Example: In auditing when examining accounts for errors; n, the sample size, is usually large. p, the error rate, is usually small.
2. Events distributed independently of one another in time:X = the number of events occurring in a fixed time interval has a Poisson distribution.Example: X = the number of telephone calls in an hour.
Probability Model:• Binomial Distribution.• Poison Distribution• Normal Distribution…….