Normalized sums of Independent Poissons and the CLT
Steven J Miller
Math 341 -- November 19  (2012)

poiss[x_,lambda_] := If[x ≥ 0,lambda^(Floor[x]) Exp[-lambda] / (Floor[x]!), 0];
normpoiss[x_,lambda_] := Sqrt[lambda]poiss[(x)Sqrt[lambda]+lambda,lambda];
stnorm[x_] := (1/Sqrt[2 Pi]) Exp[-x^2/2.];

Below is a plot of a normalized Poisson random variable with parameter lambda (if lambda is an integer we may interpret this as a sum of lambda independent standard Poisson random variables).

Manipulate[Plot[{normpoiss[x,lambda], stnorm[x]}, {x,-5,5}], {lambda,1,100}]

Below is a plot of a non-normalized Poisson random variable with parameter lambda (if lambda is an integer we may interpret this as a sum of lambda independent standard Poisson random variables).

Manipulate[Plot[{poiss[x,lambda]}, {x,0,lambda+ 5Sqrt[lambda]}], {lambda,1,100}]