1. Let X1 , X 2 , X3 , X 4 , X s be a random sample from a continuous-type distribution that is symmetric about zero. If we modify the one-sample Wilcoxon by replacing the ranks 1, 2, 3, 4, 5 by the scores 1, 1f 1, 3, 4, what is the m.g.f. of this new statistic?
2. Let X, , X 2 and Y1 , Y2 be independent random mples, each of size n = 2, from distributions with respective probability dnsity .·· functions f( x) = 1, 0 x 1, zero elsewhere, and g( y) . 3y2, 0 y 1, zero elsewhere. Compute the probability that the ranks of the Y-.values. in the combined sample of size 4 are 2 and 4.