Poisson distribution

SHREE SA’D VIDYA MANDAL

INSTITUTE OF TECHNOLOGY

DEPARTMENT OF CIVIL

ENGINEERING

Subject:-Numerical and Statistical

Methods

Topic:-Poisson Distribution

Presented by:-

Name

Arvindsai Nair

Dhaval Chavda

Shubham Yadav

Enrollment no.

130454106002

130454106001

130450106056

Poisson Distribution• The Poisson distribution was firstintroduced by Siméon DenisPoisson (1781–1840) and published,together with his probability theory, in1837 in his work “Research on theProbability of Judgments in Criminal andCivil Matters”.

•The work theorized about the number ofwrongful convictions in a given countryby focusing on certain randomvariables N that count, among otherthings, the number of discreteoccurrences (sometimes called "events"or “arrivals”) that take place during atime-interval of given length.

Siméon Denis Poisson

(1781–1840)

Poisson Distribution

In, Binomial distribution if p and n are known, it can beused But when p is small ( ) and n in very large or nis not finite,. The use of Binomial distribution is notlogical. In such situations, we use the limiting form ofbinomial distribution which is known as Poissondistribution.

Poisson Probability Distrubution

In Binomial distribution, if

(i) number of trials n is very large (i.e. )

(ii) probability of success ‘p’ is very small (i.e. ) and

(iii) average number of success np is a finite number.

i.e. np=m (m constant)

Or ( constant)

then limiting form or approximation of binomialdistribution is known as Poisson distribution.

np

n0p

1.0p

Poisson Distribution

The probability of x successes in n trails for

Poisson Distribution is given by:

!)(

x

mexP

xm

Poisson Process

Poisson process is a random

process which counts the number of

events and the time that these events

occur in a given time interval. The time

between each pair of consecutive

events has an exponential

distribution with parameter λ and each

of these inter-arrival times is assumed

to be independent of other inter-arrival

times.

Example

1. Births in a hospital occur randomly at

an average rate of 1.8 births per hour.

What is the probability of observing 4

births in a given hour at the hospital?

2. If the random variable X follows a

Poisson distribution with mean 3.4 find

P(X=6)?

Mean and Variance for

the Poisson Distribution

It’s easy to show that for this

distribution,

The Mean is:

The Variance is:

So, The Standard Deviation is:

2

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.

Properties of Poisson distribution

1. Poisson distribution is a distribution of discrete

random variable.

2. In Poisson distribution mean=variance=m. Hence

its standard deviation is .This is the acid test to

be applied to any data which might appear to

conform to Poisson distribution.

3. The sum of any finite of independent Poisson

variates is itself number a Poisson variate, with

mean equal to the sum of the means of those

variates taken separately.

m

Applications of Poisson distribution

A practical application of this

distribution was made

by Ladislaus Bortkiewicz in

1898 when he was given the

task of investigating the

number of soldiers in the

Russian army killed

accidentally by horse kicks

this experiment introduced the

Poisson distribution to the

field of reliability engineering. Ladislaus Bortkiewicz

Applications of Poisson distribution

Following are some practical applications of Poissondistribution

1. The count of - particles emitted per unit of time isuseful in analysis of any radio-active substance.

2. Number of telephone calls received at a given switchboard per small unit of time.

3. Number of deaths per day or week due to a raredisease in a big hospital

4. In industrial production to find the proportion of defectsper unit length, per unit area etc.

5. The count of bacteria per c.c. in blood

6. Distribution of number of mis-prints per page of a book