(Solution) - Let X1 Y1 Xn Yn be a random sample from a -(2025 Original AI-Free Solution)

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Let (X1, Y1),...,...,(Xn, Yn) be a random sample from a bivariate normal distribution with parameters ?x,?Y, p. We are interested in testing
H0: ?x = ?Y versus H1: ?x ? ?Y.
(a) Show that the random variables Wi = Xi - Yi are iid n(?w,?2w).
(b) Show that the above hypothesis can be tested with the statistic

Where

Furthermore, show
that, under H0, Tw ~ Student's t with n- 1 degrees of freedom. (This test is known as the paired-sample t test.)