t-distribution explorer

Four small experiments that go with the foundations page: how the tails depend on ν, why a noisy ruler fattens them, how they fall out of sphere geometry, and what they do to a loss.

1. t against the normal

Drag ν. The tail probability is the area beyond ±2.

2. Why a noisy ruler fattens the tails

20,000 experiments with n samples. Z divides by the true σ, T divides by the S estimated from the same n samples.

3. The same curve from a sphere

Random points on the sphere S^ν in ν+1 dimensions. T = √ν cot θ, with θ the angle to a fixed axis.

4. T-Loss: loss, gradient, weight

Per-pixel version. Drag ν on a log scale: ν → ∞ is MSE.

5. The weight u in an intensity model (the BayeSMM idea)

One tissue class plus a few bright pixels (think scar). Gaussian fit against a t fit whose per-pixel weights u are learned (a small EM, not the paper's full variational scheme).