N(0,1), variance as spread, why 'isotropic vs anisotropic' is the whole game in quantization.
A random variable is a number whose value is uncertain. Expectation is its average; variance is its spread. The √N law that governs sample-mean error shows up in every Monte Carlo, gradient noise, and confidence interval discussion in ML.
The Gaussian is uniquely stable under summation and uniquely flexible under linear maps. The Central Limit Theorem says sums of many independent finite-variance RVs become Gaussian regardless of where they started — which is why so much in ML is approximately normal.
A distribution is isotropic if its covariance matrix is a scalar multiple of the identity — same spread in every direction. Real ML data is wildly anisotropic. Rotation by an orthogonal matrix doesn't change the underlying distribution but does redistribute per-coordinate variance — the structural fact under TurboQuant.
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