23. 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.
24. 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.
25. Uses:
• Occurrences are independent.
• Occurrences are random.
• The probability of an occurrence is constant
over time.
26. 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 .
27. 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.
28. F o r m u l a:
• The probability that there are r occurrences in a
given interval is given by
Where,
= Mean no. of occurrences in a time interval
r =No. of trials.
30. 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
31. 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.
32. 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.
33. 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?
34. 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.
36. 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.
37. 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.
