Probability Model:

•Binomial Distribution…….•Poison Distribution•Normal Distribution.

The Binomial

Distribution…...

Defination:

Examples:

Examples::

Examples:::

Probability Model:

•Binomial Distribution.•Poison Distribution……•Normal Distribution.

POISSON DISTRIBUTION…….

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…….

The Normal Distribution…...

•The End

Thank You….