A random variable ~X is said to have the #~{Poisson distribution} with parameter &mu. _ (&mu. &in. &reals. , _ &mu. > 0 ) , _ if
P( ~X = ~x ) _ = _ fract{&mu.^~x e^{-&mu.},~x#!} , _ _ ~x >= 0
The Poisson distribution is used to model many situations which involve random events in time or space. For example, if &mu. represents the average number of events that occur in any interval of a given length of time, then the number of events that do occur in such an interval is Poisson distributed with parameter &mu..
[ Recall Taylor expansion: _ _ e ^{~a} _ = _ &sum._0^{&infty.} ~a^~x ./ ~x#! ].
The p.g.f. of the Poisson Distribution is given by
&Phi.( ~t ) _ = _ sum{fract{&mu.^~x e^{-&mu.},~x#!}~t ^~x,~x = 0,&infty.} _ = _ e^{ &mu.~t } e^{-&mu.} _ = _ e ^{&mu.( ~t - 1 )}
&Phi.'( ~t ) _ = _ &mu. e ^{&mu.( ~t - 1 )}
&Phi.''( ~t ) _ = _ &mu.^2 e ^{&mu.( ~t - 1 )}
&Phi.'( 1 ) _ = _ &mu., _ _ _ _ &Phi.''( 1 ) _ = _ &mu.^2
E( ~X ) _ = _ &mu.
var( ~X ) _ = _ &mu.^2 + &mu. - &mu.^2 _ = _ &mu.
| E( ~X ) _ = _ &mu. var( ~X ) _ = _ &mu. |