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The Poisson Distribution

Mean and Variance

Recall that the formulas for the mean and variance of a random variable are given as

Interestingly, the Poisson distribution has no upper limit and so the formulas are given as

Unfortunately, infinite sums are, in general, difficult to evaluate and the methods to determine these sums are beyond the scope of this project. However, using symbolic mathematical software (Mathcad), the following is obtained:

         and   

Amazingly, the mean and the variance of a Poisson distribution are equal.

Strong numerical evidence can also be presented for a particular Poisson distribution.

Let   and determine the finite sum to 25. Notice that both the mean and the variance are approximately 10.

x P(x) xP(x) (x-m)^2P(x)
0 4.53999E-05 0 0.004539993
1 0.000453999 0.000453999 0.036773943
2 0.002269996 0.004539993 0.145279775
3 0.007566655 0.022699965 0.370766093
4 0.018916637 0.07566655 0.680998946
5 0.037833275 0.189166374 0.94583187
6 0.063055458 0.378332748 1.008887328
7 0.090079226 0.63055458 0.810713031
8 0.112599032 0.900792257 0.450396129
9 0.125110036 1.125990321 0.125110036
10 0.125110036 1.251100357 0
11 0.113736396 1.251100357 0.113736396
12 0.09478033 1.137363961 0.37912132
13 0.072907946 0.947803301 0.656171516
14 0.052077104 0.729079462 0.833233671
15 0.03471807 0.520771044 0.867951741
16 0.021698794 0.347180696 0.781156567
17 0.012763996 0.216987935 0.625435813
18 0.007091109 0.127639962 0.453830976
19 0.003732163 0.07091109 0.302305173
20 0.001866081 0.037321626 0.186608131
21 0.00088861 0.018660813 0.107521828
22 0.000403914 0.008886101 0.058163573
23 0.000175615 0.004039137 0.029678877
24 7.31728E-05 0.001756147 0.014341863
25 2.92691E-05 0.000731728 0.00658555
       
  0.99998232 9.999530506 9.99514014