= Shot noise departures from Poisson statistics
{title2=$F_T=\operatorname{Var}(N_T)/\mathbb E[N_T]$}
The <Fano factor> $F_T$ compares count <variance> with its mean over a specified gate. For a stationary unidirectional stream with fixed charge $q$, finite count second moments, and a generalized <autocovariance> with finite integral so that the long-gate limit exists,
$$
F_\infty=\lim_{T\to\infty}\frac{\operatorname{Var}(N_T)}{\mathbb E[N_T]},\qquad
S_I^{(1)}(0^+)=2qIF_\infty.
$$
The zero-<frequency> relation concerns the continuous fluctuation spectrum, excluding the mean spectral line. It follows by writing $\operatorname{Var}(Q_T)$ as the double integral of the current <autocovariance>; for long $T$, its leading term is $T S_I^{(2)}(0)$. A finite-gate $F_T$, a frequency-dependent ratio $S_I^{(1)}(f)/(2qI)$, and a factor inferred from one filtered <variance> are not interchangeable. The last is weighted by the instrument's $|K(f)|^2$.
Two original counting counterexamples show why <electric charge> discreteness is insufficient. In $n$ independent time slots, transfer one charge with <probability> $p$ in each slot: the <binomial distribution> gives mean $np$, <variance> $np(1-p)$ and <Fano factor> $1-p$. A random common origin for the slots removes a preferred time origin; the long-gate factor remains $1-p$. Alternatively, if independent <Poisson process> bursts each transfer exactly two charges, then $N_T=2M_T$ gives $\operatorname{Var}(N_T)=2\mathbb E[N_T]$ and factor $2$. Both count individual charges; only the first model allows at most one per slot, and the second groups them in pairs.
Consequently $2qI$ is a reference for independent unidirectional arrivals, not a universal noise law for all <electrical conductors>. Correlations, charge multiplication, and bidirectional transfer require a more specific model. For example, independent opposite streams with positive current magnitudes $I_+$ and $I_-$ have net mean $I=I_+-I_-$ but one-sided fluctuation spectrum $2q(I_++I_-)$. Using $2q|I|$ would miss their fluctuations when the net current is small. At <thermal equilibrium>, the full transport model must also respect <thermal electrical noise>; adding unrelated phenomenological formulas can double count the same underlying transfers.
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