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