= Solution
<shot noise> arises from fluctuations in discrete transfer events. For equal charges $q$ arriving independently with a constant Poisson rate $\lambda$, the charge collected during $\tau$ is $Q=qN$, with
$$
\mathbb E(N)=\operatorname{Var}(N)=\lambda\tau,\quad
I=q\lambda,\quad
\operatorname{Var}(Q/\tau)=\frac{qI}{\tau}.
$$
Thus the relative counting uncertainty is $1/\sqrt{\lambda\tau}$. Longer integration improves the precision of the mean current even when the microscopic arrivals remain irregular.
In the <frequency> range where independent arrivals can be treated as instantaneous pulses, the one-sided current <power spectral density> is
$$
\boxed{S_I^{(1)}(f)=2qI\quad\mathrm{A^2\,Hz^{-1}}}.
$$
For an ideal unit-gain rectangular bandwidth $B$, this gives $i_{\rm rms}=\sqrt{2qIB}$. <boxcar averaging of Poisson shot noise> over duration $\tau$ has <effective noise bandwidth> $B_{\rm n}=1/(2\tau)$, reproducing the counting result $i_{\rm rms}=\sqrt{qI/\tau}$. The two-sided spectral convention instead assigns $qI$ to each sign of <frequency>. Stating that convention prevents a factor-of-two error.
A <photodiode> illuminated at a stable rate provides a useful measurement: subtract the mean current, amplify its fluctuations through a calibrated <linear filter>, and compare the <variance> with the mean <photocurrent>. Independent detected photoelectrons predict a <variance> proportional to mean current. Measure the dark and amplifier backgrounds separately, check the filter's <effective noise bandwidth> and vary illumination to distinguish this term from classical intensity noise, whose <variance> can scale quadratically with mean current. The <shot noise> model concerns counting statistics, whereas <thermal electrical noise> can exist in an unbiased <resistor>. <Walter Schottky>'s motivating analysis treated amplifier limits and independent <electron> transfers in thermionic discharge tubes.<Über spontane Stromschwankungen in verschiedenen Elektrizitätsleitern by Walter Schottky>[Part II, especially pp. 557–563]{ref}
The Poisson result is a model, not a universal consequence of charge quantization. Correlated arrivals, <detector dead time>, avalanche multiplication or a non-Poisson light source can change the fluctuations. Filtering reduces integrated noise at the cost of temporal resolution. Increasing the detected event rate reduces relative <shot noise> but raises its absolute <variance>; <balanced photodetection> cancels shared classical fluctuations, not the independent <shot noise> of two uncorrelated detectors.
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