= Boxcar averaging of Poisson shot noise
{title2=$\operatorname{Var}(\overline I_\tau)=qI/\tau$}
Independent arrivals of charges $q$ at rate $\lambda$ give a <Poisson distribution> for the number $N$ in time $\tau$, with mean and <variance> $\lambda\tau$. Since $\overline I_\tau=qN/\tau$ and $I=q\lambda$, its <variance> is $qI/\tau$. This agrees with integrating the one-sided <shot noise> <power spectral density> $2qI$ through the averaging filter: its <effective noise bandwidth> is $1/(2\tau)$. A two-sided convention instead assigns density $qI$ to each sign of <frequency>.
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