Packet 15: Bonus 2
If a matrix has this property, the behavior of its eigenvalues is governed by the Marchenko-Pastur theorem and the Tracy-Widom law. For 10 points each:
[10m] Name this property of matrices that Eugene Wigner introduced to physics to model the energy levels of heavy atomic nuclei. Matrices with this property have an empirical spectral measure and include the sample covariance.
ANSWER: random [or word forms like randomness; accept stochastic]
[10h] For a random symmetric matrix with IID entries, the limiting density of its rescaled eigenvalues has this shape. That statement named for this shape is arguably the most fundamental result in random matrix theory.
ANSWER: semicircle [accept semicircular; accept Wigner’s semicircle distribution; accept semi-ellipse; reject “circle” or “ellipse” or word forms]
[10e] Wishart’s distribution on random matrices is used to derive Hotelling’s multivariate generalization of this test. This test is used for the difference of means of two groups with unknown variances, unlike the similar z-test.
ANSWER: Student’s t-test [accept Hotelling’s T-squared test]
<TM, Other Science (Math)> | NAFTA-Packet-Extras
| Heard | PPB | E % | M % | H % |
|---|---|---|---|---|
| 2 | 20.00 | 100% | 50% | 50% |
Conversion
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Summary
| Tournament | Edition | Match | Heard | PPB | E % | M % | H % |
|---|---|---|---|---|---|---|---|
| 2025 NAFTA at Toronto | 09/13/2025 | ✓ | 1 | 10.00 | 100% | 0% | 0% |
| 2025 NAFTA at Toronto | 09/13/2025 | ✓ | 1 | 30.00 | 100% | 100% | 100% |