= Solution
<Johnson-Nyquist noise> is equilibrium electrical fluctuation caused by thermal agitation of charge carriers. A <resistor> can generate a fluctuating terminal <voltage> even when its mean current is zero. In the classical <frequency> range $hf\ll k_BT$, the <Fluctuation-dissipation theorem> relates this <noise> to dissipative resistance. The one-sided <voltage> <power spectral density> of an unloaded <resistor> is
$$
\boxed{S_V^{(1)}(f)=4k_BTR\quad\mathrm{V^2/Hz}.}
$$
Equivalently its short-circuit current spectrum is $4k_BT/R$. Over a flat bandwidth $B$ the rms <voltage> is $\sqrt{4k_BTRB}$. A real measurement instead integrates $S_V(f)|H(f)|^2$ over positive <frequency>, with loading and the filter's <effective noise bandwidth> included. A flat spectrum does not imply an infinite physically measurable <variance>; the circuit and the quantum high-frequency regime limit the ideal white-noise model.
Cooling or narrowing the accepted bandwidth reduces <thermal noise>. Reducing resistance lowers its open-circuit <voltage> <noise> but raises its short-circuit current <noise>, so the appropriate choice depends on source impedance and amplifier loading. Thermal agitation is not a fluctuating externally applied <voltage> that can be removed by perfect shielding alone.
<Shot noise> arises from discrete transfer events. In the simplest <shot noise counting model>, equal charges $q$ arrive independently at constant rate $r$. A gate of duration $t$ contains a Poisson number $K$, with mean and <variance> $rt$. Therefore mean current is $I=qr$, and the averaged-current <variance> is $qI/t$. The corresponding one-sided low-frequency spectrum is
$$
\boxed{S_I^{(1)}(f)=2qI,\qquad I_{\mathrm{rms,noise}}=\sqrt{2qIB}}
$$
for a flat accepted bandwidth. <Electron> current has $q=e$. Vacuum emission or photodetection can approximate independent arrivals; correlations in a conductor can suppress or enhance the fluctuations. Discreteness by itself does not guarantee Poisson statistics.
For a fixed mean current the ideal shot-noise formula is not proportional to <temperature>, whereas Johnson <noise> is. Increasing current raises absolute <shot noise> but lowers its fractional size, since $\sqrt{2qIB}/I\propto I^{-1/2}$. Both can appear approximately white within a useful <frequency> band, and independent <noise> powers add after referring them to a common measurement point. Their spectra must use the same convention: two-sided densities are half these one-sided values. A real low-current experiment can also contain amplifier <noise>, low-frequency drift and correlated transfer <noise>, so a mean current or resistance alone does not specify its complete <noise> budget.
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