The number of television sets sold at a department store during a given week. .... A sales firm receives on average three calls per hour on its toll-free number.
POISSON DISTRIBUTION
DR.HAYDER ABBAS DREBEE
POISSON DISTRIBUTION
Poisson Distribution
Derived from the French mathematician Simeon D. Poisson.
A discrete probability distribution that is useful when n is large and p is small and when the independent variables occur over a period of time or given area or volume.
Conditional to apply the Poisson Probability Distribution
x is a discrete random variable
The occurrences are random
The occurrences are independent
The following examples show the application of the Poisson probability distribution.
The number of accidents that occur on a highway given during a one-week period.
The number of customers entering a grocery store during a one-hour interval.
The number of television sets sold at a department store during a given week.
The number of typing errors per page.
A certain type of fabric made contains an average 0.5 defects per 500 yards.
Calculating Poisson Probability a) Using Poisson Probability Distribution Formula According to the Poisson probability distribution, the probability of x occurrences in an interval is:
e x P(X)= x! Where; : mean number of occurrences in that interval e : approximately 2.7183
POISSION DISTRIBUTION
DR.HAYDER ABBAS DREBEE
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POISSON DISTRIBUTION
DR.HAYDER ABBAS DREBEE
Example Find probability P(X; ), using the Poisson formula for P(5;4)
Solution: P(X=5) =
e 4 (45 ) 5!
= 0.1563
Example If there are 200 typographical errors randomly distributed in a 500-page manuscript, find the probability that a given page contains exactly three errors.
Solution: Find the number of errors: = 200/500 = 0.4 (error per page)
P(X=3)=
e 0.4 0.43 3!
= 0.0072
POISSION DISTRIBUTION
DR.HAYDER ABBAS DREBEE
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POISSON DISTRIBUTION
DR.HAYDER ABBAS DREBEE
b) Using the Table of Poisson Probabilities
The probabilities for a Poisson distribution can also be read from the table of Poisson probabilities.
How to read the probability value from the table of Poisson probabilities Determining P (x 3) for = 1.5 The table of cumulative Poisson probabilities (less than)
e k P( X x) k! k 0 x
= 1.5
x
1.1
1.2
1.3
1.4
1.5
1.6
1.7
0
0.3329
0.3012
0.2725
0.2466
0.2231
0.2019
0.1827
1 2
0.6990
0.6626
0.6268
0.5918
0.5578
0.5249
0.4932
0.9004
0.8795
0.8571
0.8335
0.7834
0.7572
3 4
0.9743
0.9662
0.9569
0.9463
0.8088 0.9344
0.9212
0.9068
0.9946
0.9923
0.9893
0.9857
0.9814
0.9763
0.9704
5
0.9990
0.9985
0.9978
0.9968
0.9955
0.9940
0.9920
6
0.9999
0.9997
0.9996
0.9994
0.9991
0.9987
0.9981
7
1.0000
1.0000
0.9999
0.9999
0.9998
0.9997
0.9996
x3
P (x 3) = 0.9344
If you are using cumulative Poisson probabilities table; P(X ≤ x); it is easier to calculate various from of binomial distribution such as: Equally
P(X = x) = P(X ≤ x) - P(X ≤ x - 1) or using Poisson formula
