Poisson sample-mean rate function
= Poisson sample-mean rate function
{c}
{title2=$I_P(x)=x\log(x/\lambda)-x+\lambda\quad(x\ge0)$}
For sample means of <Poisson random variables> of mean $\lambda>0$, the <cumulant-generating function> is $\lambda(e^\theta-1)$. Its <Legendre-Fenchel transform> is the displayed <good rate function>, with $0\log0=0$, and is infinite for $x<0$. Near its zero, $I_P(\lambda+h)=h^2/(2\lambda)+O(h^3)$. This local quadratic explains the <Poisson moderate deviation principle>.