Open Study Questions

What is a short simple proof of the Cauchy-Schwarz inequality?
For any stationary markov process, is the autocorrelation of an interval the product of the autocorrelations of subintervals?
For a multivariate distribution, is the sum of sample variances a better estimator than the sum of population variances?
For a Bernoulli distribution, is sample variance a better estimator than simply the definition of variance?
What is the relationship between the variance and entropy of independent one-hot vectors?
What is an extension of Shannon entropy to continuous variables?