A friend asked how to tell whether a biomarker can predict cardiotoxicity. I build ROC curves by hand on synthetic case and control data, then use them to assess the diagnostic power of simulated dose-response curves.
Deriving the delta method from a Taylor expansion and simulating its use for variance stabilizing transformations of Poisson and exponential data and for standard errors in polynomial regression.
Using the squared Mahalanobis distance, which is chi-squared distributed under multivariate normality, to compare connectivity fingerprints between one cortical region and every other region, with Ledoit-Wolf covariance estimates and a check that the empirical distances follow the expected distribution.
Working through the definition of convergence in probability from Casella and Berger, using Chebyshev's inequality to show the weak law of large numbers for the sample mean and variance, and simulating normal samples to watch both estimates converge.
A sum of independent Poisson variables is Poisson with the summed rate. Conditioned on that sum, the individual counts follow a multinomial distribution.
Deriving the marginal and conditional distributions of a partitioned multivariate normal.
Deriving each block of the inverse of a 2x2 partitioned matrix using the Schur complements of A and D. I kept running into these identities in a linear models course, including on a midterm.