= Gaussian sign-correlation identity
{c}
{title2=$\mathbb E[\operatorname{sign}G\operatorname{sign}H]=\frac2\pi\arcsin\rho$}
For centered, unit-variance jointly <Gaussian random variables> with <correlation coefficient> $\rho$, the displayed identity holds. Realize the pair as projections onto unit <vectors> of angle $\theta=\arccos\rho$. Rotational symmetry of the two-dimensional <Gaussian random vector> direction makes sign disagreement have <probability> $\theta/\pi$. The <expectation> is therefore $1-2\theta/\pi=(2/\pi)\arcsin\rho$. Perfectly correlated and anticorrelated endpoints follow directly; the sign at zero does not affect the result.
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