POISION DISTRIBUTIONPresented By:-

1. Shubham Ranjan2. Siddharth Anand

IntroductionThe distribution was first introduced by

Simon Denis Poisson (1781–1840) and published, together with his probability theory, in 1837 in his work Recherches

sur la probabilité des jugements en matière criminelle et en matière civile

(“Research on the Probability of Judgments in Criminal and Civil

Matters”).

Definition The Poisson distribution is a probability

model which can be used to find the probability of a single event occurring a given number of times in an interval of (usually) time. The occurrence of these events must be determined by chance alone which implies that information

about the occurrence of any one event cannot be used to predict the

occurrence of any other event.

The Poisson ProbabilityIf X is the random variable then ‘number of occurrences in a given interval ’for which the

average rate of occurrence is λ then, according to the Poisson model, the

probability of r occurrences in that interval is given by

P(X = r) = e−λλr /r ! Where r = 0, 1, 2, 3, . . .NOTE : e is a mathematical constant.

e=2.718282 and λ is the parameter of the distribution. We say X follows a Poisson

distribution with parameter λ.

The Distribution arise When the Event being Counted occur

• Independently• Probability such that two or more event

occur simultaneously is zero• Randomly in time and space• Uniformly (no. of event is directly

proportional to length of interval).

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.

Types of Poisson Process

• Homogeneous• Non-homogeneous• Spatial• Space-time

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)?

The Shape of Poisson Distribution

• Unimodal• Exhibit positive skew (that

decreases a λ increases)• Centered roughly on λ• The variance (spread) increases as λ

increases

Mean and Variance for thePoisson Distribution

• It’s easy to show that for this distribution,

The Mean is:

• Also, it’s easy to show that

The Variance is:

So, The Standard Deviation is:

2

Properties• The mean and variance are both equal to

.

• The sum of independent Poisson variables is a further Poisson variable with mean equal to the sum of the individual means.

• The Poisson distribution provides an approximation for the Binomial distribution.

Sum of two Poisson variables

Now suppose we know that in hospital A births occur randomly at an average rate of 2.3 births per hour and in hospital B

births occur randomly at an average rate of 3.1 births per hour. What is the

probability that we observe 7 births in total from the two hospitals in a given 1

hour period?

Comparison of Binomial & Poisson Distributions with Mean μ = 1

0

0.1

0.2

0.3

0.4

0.5

Prob

abilit

y

0 1 2 3 4 5m

poissonbinomial

N=3, p=1/3

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

Prob

abili

ty

0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0

m

binomialpoisson

N=10,p=0.1

Clearly, there is not much difference

between them!

For N Large & m Fixed:Binomial Poisson

Approximation

If n is large and p is small, then the Binomial distribution with

parameters n and p is well approximated by the Poisson

distribution with parameter np, i.e. by the Poisson distribution

with the same mean

Example

• Binomial situation, n= 100, p=0.075

• Calculate the probability of fewer than 10 successes.

pbinom(9,100,0.075)[1] 0.7832687 This would have been very tricky

with manual calculation as the factorials are very large and the

probabilities very small

• The Poisson approximation to the Binomial states that will be equal to np, i.e. 100 x 0.075

• so =7.5• ppois(9,7.5)[1] 0.7764076• So it is correct to 2 decimal

places. Manually, this would have been much simpler to do than the Binomial.

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