Homework
Standing homework instructions
You may disregard measure-theoretic niceties about conditioning on measure-zero sets, almost-sure equality vs. actual equality, “all functions” vs. “all measurable functions,” etc. (unless the problem is explicitly asking about such issues).
If you need to write code to answer a question, show your code. If you need to include a plot, make sure the plot is readable, with appropriate axis labels and a legend if necessary. Points will be deducted for very hard-to-read code or plots.
Anytime I ask you to calculate things numerically, it is implicit that Monte Carlo integration (calculating an expectation by repeatedly sampling data from an appropriate distribution and taking the average) is a valid numerical method. There is no need to ask permission, but please use good judgment about how many samples to take so that your numerical error is not too high: if you report a number it should be correct to a few significant digits; if you are comparing two numbers, the precision of your calculation should be high enough for the difference to be meaningful; and plots of smooth functions should appear smooth enough for the plot to be readable.
Unless otherwise stated, assume asymptotic limits are taken as \(n\to \infty\). If I ask for a “limiting distribution,” I mean do an appropriate centering and scaling to find a limiting distribution that is non-degenerate (not converging in probability to a constant). That is, find sequences \(a_n\) and \(b_n\) such that \(b_n(X_n−a_n)\) converges to a non-degenerate limiting distribution.