The Narrowest Window for a Third Correlation

Worked solution

Standardise the variables and treat them as unit vectors in a Hilbert space; correlation is the cosine of the angle. The angle between $X$ and $Y$ is $\theta$ with $\cos\theta = 4/5$, likewise for $Y,Z$. The angle between $X$ and $Z$ is at most $2\theta$, so the smallest cosine is $\cos 2\theta = 2(16/25) - 1 = 7/25$. Equivalently, the correlation matrix must be positive semidefinite, which forces $r_{XZ} \ge r_{XY}r_{YZ} - \sqrt{(1-r_{XY}^2)(1-r_{YZ}^2)} = 16/25 - 9/25 = 7/25$. Attained when $X$ and $Z$ lie on opposite sides of $Y$ in the plane.