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

d'T Plot- versus T-T, and using your knowledge of curve fitting find the value of dt the constant k. Write and implement necessary code. dT Hint: Model == kTkT is analogous to y-ax+b dt

Derive and explain briefly the weight update mechanism in Widrow-Hoff learning rule using gradient descent algorithm and least squared error.