Lambdia

Correlation and dependence

2 articles

Slide It, Stretch It, and the Correlation Does Not Move

A shift leaves every deviation from the mean untouched, and a stretch multiplies the covariance and one standard deviation by the same factor, so both cancel out of the ratio. The tempting answer of five times rho is worse than wrong: at rho = 0.40 it names 2.0, which Cauchy-Schwarz forbids any correlation from reaching. The article carries the general affine rule, the sign flip a negative factor produces, and the curved maps the invariance does not survive.

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Two Stocks at 0.7 With the Same Index, and Still −0.02 With Each Other

The two correlations you are handed do not pin the third one down, but they fence it into exactly [−1/50, 1], and that interval dips below zero. The fence falls out of a 3×3 determinant read as a quadratic in the unknown, and out of a picture: 0.7 is an angle of 45.573°, both stocks live on a cone of that half-angle around the index, and putting them on opposite sides opens 91.146° between them. Also here: why the real tipping point is ab ≥ 0 together with a² + b² ≥ 1 rather than "both above 0.707", why 0.9 and 0.5 force a positive answer while 0.9 and −0.9 allow −1, why standing on the floor costs a rank, and why three Bernoulli(0.5) indicators with the same two correlations are confined to [0.40, 1] instead.

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