2. Introduction
The 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”).
3. 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.
4. The Poisson Probability
If 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.
5. 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).
6. 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.
7. Types of Poisson Process
• Homogeneous
• Non-homogeneous
• Spatial
• Space-time
8. 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)?
9. 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
10. Mean and Variance for the
Poisson 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
11. 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.
12. 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?
13. Comparison of Binomial & Poisson Distributions
with Mean μ = 1
0
0.1
0.2
0.3
0.4
0.5
Probability
0 1 2 3 4 5m
poisson
binomial
N=3, p=1/3
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
Probability
0.0 1.0 2.0 3.0 4.0 5.0 6.0 7.0
m
binomial
poisson
N=10,p=0.1
Clearly, there is not much difference
between them!
For N Large & m Fixed:
Binomial Poisson
14. 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
15. 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
16. • 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.
