Mathematical Methods for Physics and Engineering - Matematica.NET

30.8 IMPORTANT DISCRETE DISTRIBUTIONS0.3f(x)f(x)λ =1 λ =20.30.20.20.1 0.10001 2 3 4 5 x0 1 2 3 4 5 67xf(x)0.3λ =50.20.1001234 567 8 9 10 11xFigure 30.12 Three Poisson distributions for different values of the parameterλ.The above example illustrates the point that a Poisson distribution typicallyrises and then falls. It either has a maximum when x is equal to the integer partof λ or, if λ happens to be an integer, has equal maximal values at x = λ − 1andx = λ. The Poisson distribution always has a long ‘tail’ towards higher values of Xbut the higher the value of the mean the more symmetric the distribution becomes.Typical Poisson distributions are shown in figure 30.12. Using the definitions ofmean and variance, we may show that, for the Poisson distribution, E[X] =λ andV [X] =λ. Nevertheless, as in the case of the binomial distribution, performingthe relevant summations directly is rather tiresome, and these results are muchmore easily proved using the MGF.The moment generating function for the Poisson distributionThe MGF of the Poisson distribution is given byM X (t) =E [ e tX] =∞∑ e tx e −λ λ x ∑∞= e −λx!x=01177x=0(λe t ) xx!= e −λ e λet = e λ(et −1)(30.104)