= Solution
Normalize each reporter by its mean intensity, after correcting background and unequal brightness, so equal expression lies on $y=x$. Write $x=1+e+i_C$, $y=1+e+i_Y$, where the common fluctuation $e$ is <extrinsic gene-expression noise> and the independent reporter-specific fluctuations $i_C,i_Y$ are <intrinsic gene-expression noise>. The <extrinsic gene-expression noise> includes shared variation in cell size, expression machinery and growth state; <intrinsic gene-expression noise> includes independent molecular production and removal events at each reporter.
If <intrinsic gene-expression noise> is much smaller, points form a narrow elongated cloud along $y=x$: a cell that is bright in cyan is usually bright in yellow. If the two contributions have comparable magnitude, the cloud is still positively correlated but has substantial width perpendicular to $y=x$, because the two reporters can differ within the same cell. With <variance> $\sigma_e^2$ in $e$ and equal <variance> $\sigma_i^2$ in each independent intrinsic term,
$$
\operatorname{Cov}(x,y)=\sigma_e^2,\qquad
\operatorname{Var}(y-x)=2\sigma_i^2,\qquad
\operatorname{Corr}(x,y)=\frac{\sigma_e^2}{\sigma_e^2+\sigma_i^2}.
$$
Equal intrinsic and extrinsic <variances> therefore give <correlation coefficient> $1/2$, rather than an uncorrelated circular cloud. For mean-normalized reporters, $\eta_{\rm int}^2=\langle(x-y)^2\rangle/2$ and $\eta_{\rm ext}^2=\langle xy\rangle-1$. Measurement noise must be separately estimated before assigning perpendicular scatter to expression itself. These are the geometric diagnostics of the original dual-reporter experiment.<Stochastic Gene Expression in a Single Cell by Michael B. Elowitz and others>[Figure 1 and definition of intrinsic and extrinsic noise: within-cell reporter differences versus correlated cell-to-cell variation]{ref}
The lower panels of <2/c> show these two cases. Repeated observations of one cell can move both along and across the diagonal; a static population cloud alone does not determine the time scales of the fluctuations.
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