= Solution
For constant positive <reaction rates>, the <integrating factor> for the <RNA> equation gives
$$
\boxed{R(t)=R_*+[R(0)-R_*]e^{-\gamma_Rt},\qquad
R_*=\frac{k_R}{\gamma_R},\qquad P_*=\frac{k_PR_*}{\gamma_P}.}
$$
For completeness, the <protein> transient when $\gamma_P\ne\gamma_R$ is
$$
P(t)=P_*+[P(0)-P_*]e^{-\gamma_Pt}
+k_P[R(0)-R_*]\frac{e^{-\gamma_Rt}-e^{-\gamma_Pt}}{\gamma_P-\gamma_R}.
$$
At equal degradation rates, the last term becomes $k_P[R(0)-R_*]t e^{-\gamma_Pt}$, its continuous limit.
One transcript per minute means $k_R=1\,\mathrm{min}^{-1}$ for a single active <gene>. A <mean> <RNA> lifetime of ten minutes means $\gamma_R=0.1\,\mathrm{min}^{-1}$, not a ten-minute half-life. The steady <mean> is therefore $\boxed{\langle R\rangle=10}$ <RNA> molecules per cell.
For the Fano factors below, $R,P$ denote molecule numbers in a fixed cell. If they instead denote dimensional concentrations, the same deterministic equations hold after rescaling, but the conversion to molecule-number <variance> introduces the cell-volume factor; <variance> divided by concentration is not intrinsically a dimensionless count <Fano factor>.
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