= Distribution of the maximum of correlated receiver horizons
The <distribution of the maximum of correlated receiver horizons> must use the joint timing law. Let $H=\max(H_1,H_2)$. Then $\Pr(H\le h)=\Pr(H_1\le h,H_2\le h)$, which generally is not the product of the marginal probabilities. The independent special case is derived in <channel switching under uncertainty in a combined receiver horizon>.
For an explicit correlated model, suppose $(H_1,H_2)$ has a <bivariate normal distribution> with common <mean> $\mu$, common <standard deviation> $s>0$, and <correlation coefficient> $-1<\rho<1$. With $z=(h-\mu)/s$, the <cumulative distribution function> and density of the maximum are
$$
F_H(h)=\Phi_2(z,z;\rho),\qquad
f_H(h)=\frac2s\phi(z)\Phi\!\left(z\sqrt{\frac{1-\rho}{1+\rho}}\right).
$$
Here $\phi,\Phi$ are the standard <normal distribution> density and cdf, and $\Phi_2$ is the bivariate cdf. The density follows by differentiating along both boundaries and using the conditional normal law. The official SciPy reference supplies a cdf method parameterized by the <mean> vector and <covariance matrix>; it is a numerical evaluation, not an assumption of channel independence.<SciPy 1.16.2 documentation>[scipy.stats.multivariate_normal, Parameters: mean and cov; Methods: cdf, maxpts, abseps, releps]{ref}
The equal-variance model also gives exact moments. Write $H=(S+|D|)/2$, with $S=H_1+H_2$ and $D=H_1-H_2$. They are independent normal variables, with $\operatorname{Var}(D)=2s^2(1-\rho)$. The absolute-normal moment then gives
$$
EH=\mu+s\sqrt{\frac{1-\rho}{\pi}},\qquad
\operatorname{Var}(H)=s^2\left(1-\frac{1-\rho}{\pi}\right).
$$
For an original shared-clock example, let $\mu=100\,\mathrm{ns}$, $s=1\,\mathrm{ns}$, and $\rho=1/2$. Then $EH\approx100.399\,\mathrm{ns}$ and the <standard deviation> is about $0.917\,\mathrm{ns}$. Positive correlation reduces the expected gain from selecting the larger horizon, while increasing its <variance> relative to the independent equal-channel case.
A <common-clock contribution to a maximum receiver horizon> can be modeled for nonnegative $\rho$ by $H_j=\mu+s(\sqrt\rho Z_0+\sqrt{1-\rho}Z_j)$ with independent standard normal inputs. As $\rho$ tends to one, both channels share the same uncertain horizon: selection does not remove the common error. Actual receiver timing correlations must be justified by calibration and reference reuse; the example neither measures them nor guarantees that the receiver is invertible.
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