Use acceptance-rejection to simulate n = 1000 from the standard Cauchy distribution. Please indicate clearly which distribution you are sampling from and from that, you must determine the optimal c value and show all work and code. You can compare your answers with the AR package results. Plot the sample in a histogram in R and overlay the true Cauchy density superimposed for comparison.

To examine whether there is any association between the performances on the two tests, T1 and T2 (test T1 was conducted before T2), and whether T2 scores could be predicted in terms of T1 scores. A random sample of 150 students in Statistics major was selected. Let the scores be denoted S1 and S2 for tests T1 and T2, respectively. The tests had different maximum points. The following outputs using R codes were obtained. fit[t]) (Intercept) 0.10152 2.51003 0.04 0.968 S1 0.26723 0.01921 13.91 <2e-16 *** Signif. codes: O ***' 0.001 **' 0.01 (*' 0.05 '.' 0.1 '? 1 Residual standard error: 4.472 on 148 degrees of freedom Multiple R-squared: F-statistic: 193.6 on 1 and 148 DF, 0.5667, Adjusted R-squared: 0.5638 p-value: < 2.2e-16 plot (fit$fitted,fit$residuals, xlab="fitted values", ylab="residuals") qqnorm (fit$residuals) Normal Q-Q Plot oo 00 00 00 O Go a god 00 00 0. of 25 30 35 40 45 -2 2 fitted values Theoretical Quantiles esiduals 15 Score: Sample Quantiles 091

Notice that Ro1(h) can be interpreted as the misclassification rate. That is, if R01(h) .7, then predicting h would result in the wrong answer for 70% of the data points. Given the data set {4, 2, 4, 1, 3, 4, 4, 3, 2, 5}, plot the empirical risk Ro1(h) for h = [0, 5]. Hint: the function should have point discontinuities. = c) Is gradient descent useful for minimizing the risk with zero-one loss? Why or why not? Make reference to your plot of the risk in your answer. Hint: the risk is indeed non-convex, but gradient descent can still be useful for minimizing non-convex functions. Is there some other reason?