The document discusses the Poisson experiment, which is used to model rare, random events that occur independently within intervals of a specified metric like time, distance, or volume. Some key properties are that the probability of an event is the same across equal intervals and independent between intervals. Examples given include accidents, bacteria, or computer viruses. The document provides an example of calculating probabilities of certain virus counts using the Poisson distribution and defines that for a Poisson random variable, the expected value and variance are equal to the mean.

1. 1

2. The Poisson Experiment
• Used to find the probability of a rare event
• Randomly occurring in a specified interval
– Time
– Distance
– Volume
• Measure number of rare events occurred in the
specified interval
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3. The Poisson Experiment - Properties
• Counting the number of times a success occur in
an interval
• Probability of success the same for all intervals
of equal size
• Number of successes in interval independent of
number of successes in other intervals
• Probability of success is proportional to the size
of the interval
• Intervals do not overlap
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4. The Poisson Experiment - Properties
• Some of the properties for a Poisson experiment
can be hard to identify.
• Generally only have to consider whether:-
– The experiment is concerned with counting number of
RARE events occur in a specified interval
– These RARE events occur randomly
– Intervals within which RARE event occurs are
independent and do not overlap
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5. The Poisson Experiment
Typical cases where the Poisson experiment
applies:
– Accidents in a day in Soweto
– Bacteria in a liter of water
– Dents per square meter on the body of a car
– Viruses on a computer per week
– Complaints of mishandled baggage per 1000
passengers
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6. The Poisson Experiment
Calculating the Poisson Probability
Determining x successes in the interval:
 e
x 
P( X  x)  P( x) 
x!
where,  = mean number of successes in interval
e = base of natural logarithm  2.71828
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7. The Poisson Experiment - Example
• On average the anti-virus program detects 2 viruses
per week on a notebook
Are the conditions required for the Poisson experiment met?
• Time interval of one week
• μ = 2 per week
• Occurrence of viruses are independent
• Can calculate the probabilities of a certain number of viruses
in the interval
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8. The Poisson Experiment - Example
• Let X be the Poisson random variable indicating the
number of viruses found on a notebook per week
Calculate the probability that threeviruses occur
one viruses occur
zerovirus occur
 x e 
P ( X  x)  P( x) 
x!
e2 20
P( X  0)  P(0)  0!
 0.1353
e2 21
P( X  1)  P(1)  1!
 0.2707
e2 23
P( X  3)  P(3)  3!
 0.1804
e2 27
P( X  7)  P(7)   0.0034 8
. 7!
.

9. The Poisson Experiment - Example
• Let X be the Poisson random variable indicating the
number of viruses found on a notebook per week
X P(X)
e2 20
P( X  0)  P(0)  0!
 0.1353 0 0.1353
1 0.2707
e2 21
P( X  1)  P(1)  1!
 0.2707 2 0.2707
3 0.1804
e2 23
P( X  3)  P(3)  3!
 0.1804 ↓ ↓
7 0.0034
e2 27
P( X  7)  P(7)   0.0034 ↓ ↓
. 7!
.
. ∑P(X) ≈ 19

10. The Poisson Experiment - Example
• Calculate the probability that two or less than two
viruses will be found per week
X P(X)
P(X ≤ 2) 0 0.1353
1 0.2707
= P(X = 0) + P(X = 1) + P(X = 2) 2 0.2707
3 0.1804
= 0.1353 + 0.2707 + 0.2707
↓ ↓
= 0.6767 7 0.0034
↓ ↓
∑P(X) ≈ 10
1

11. The Poisson Experiment - Example
• Calculate the probability that less than three
viruses will be found per week
X P(X)
P(X < 3) 0 0.1353
1 0.2707
= P(X = 0) + P(X = 1) + P(X = 2) 2 0.2707
3 0.1804
= 0.1353 + 0.2707 + 0.2707
↓ ↓
= 0.6767 7 0.0034
↓ ↓
∑P(X) ≈ 11
1

12. The Poisson Experiment - Example
• Calculate the probability that more than three
viruses will be found per week
X P(X)
P(X > 3) 0 0.1353
1 0.2707
= P(X = 4) + P(X = 5) + ……… 2 0.2707
3 0.1804
= 1 – P(X ≤ 3)
↓ ↓
= 1 - 0.8571 7 0.0034
↓ ↓
=0.1429
∑P(X) ≈ 12
1

13. The Poisson Experiment - Example
• Calculate the probability that four viruses will be
found in four weeks
• μ = 2 x 4 = 8 in four weeks
P ( X  4)
e 8 84

4!
 0.0573
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14. The Poisson Experiment - Example
• Calculate the probability that two or less than two
viruses will be found in two weeks
• μ = 2 x 2 = 4 in two weeks
P( X  2)
 P( X  0)  P( X  1)  P( X  2)
4 0 4 1 4 2
e 4 e 4 e 4
  
0! 1! 2!
 0.0183  0.0733  0.2381
 0.2381 14

15. The Poisson Experiment
• Mean and standard deviation of Poisson
random variable
  E( X )  
    Var ( X )  
2
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16. The Poisson Experiment - Example
– What is the expected number of viruses on the
notebook per week?
  E( X )    2
– What is the standard deviation for the number of
viruses on the notebook per week?
    Var ( X )    2  1.41
2
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