Page 204 - DMTH404_STATISTICS

Statistics

Notes 14.6.5 Fitting of a Poisson Distribution

To fit a Poisson distribution to a given frequency distribution, we first compute its mean m.
Then the probabilities of various values of the random variable r are computed by using the
e  m .m r
probability mass function ( ) r = . These probabilities are then multiplied by N, the total
P
! r
frequency, to get expected frequencies.

Example 23:

The following mistakes per page were observed in a book :
No . of mistakes per page : 0 1 2 3

Frequency : 211 90 19 5
Fit a Poisson distribution to find the theoretical frequencies.

Solution.
The mean of the given frequency distribution is

0 211 1 90 2 19 3 5 143
+
+
´
´
+
´
´
m = = = 0.44
+
+
211 90 19 5 325
+
Calculation of theoretical (or expected) frequencies
e  0.44 (0.44 ) r
P
We can write ( ) r = . Substituting r = 0, 1, 2 and 3, we get the probabilities for
! r
various values of r, as shown in the following table.
Expected Frequencies Approximated
r P ( ) r N P ( ) r
´
to the nearest integer
0 0.6440 209.30 210
1 0.2834 92.10 92
2 0.0623 20.25 20
3 0.0091 2.96 3
Total 325

14.6.6 Features of Poisson Distribution

(i) It is discrete probability distribution.
(ii) It has only one parameter m.

(iii) The range of the random variable is 0  r < .
(iv) The Poisson distribution is a positively skewed distribution. The skewness decreases as m
increases.

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