Consider a deck consisting of seven cards, marked 1, 2, . . . , 7. Three of these cards are selected at random. Define an rv W by W = the sum of the resulting numbers, and compute the pmf of W. Then compute m and s2. [Hint: Consider outcomes as unordered, so that (1, 3, 7) and (3, 1, 7) are not different outcomes. Then there are 35 outcomes, and they can be listed. (This type of rv actually arises in connection with a statistical procedure called Wilcoxon’s rank-sum test, in which there is an x sample and a y sample and W is the sum of the ranks of the x’s in the combined sample; see Section 15.2.)

Answer: Step1: Consider a deck consisting of seven cards, marked 1,2,3,4,5,6,7. Three of these cards are selected at random. c = 35 ways. 3 Thus the total number of outcomes is 35. i.e, (1,2,3),(1,2,4),(1,2,5),(1,2,6),(1,2,7),.......,(5,6,7). Then, P(each outcome) = 35 Here, W = sum of the resulting numbers. i.e, (1+2+3 = 6, 1+2+4 = 7 ….) Therefore, W = (6,7,8,9,10,...........,18) Since there is just one outcome with W value 6. P(6) = P(W = 6) = 1 31 P(7) = P(W = 7) = 35 Similarly there are 3 outcomes with W value 9 [(1,2,6) (1,3,5) (2,3,4)] So P(9) = 3 . 35 Step2: Continuing in this manner yields the following distribution. W 6 7 8 9 10 11 12 13 14 15 16 17 18 1 1 2 3 4 4 5 4 4 3 2 1 1 P(w) 35 35 35 35 35 35 35 35 35 35 35 35 35 Since the distribution is symmetric about W = 12. So = 12. 2 18 2 And S = (w 12) p(w) w=0 S = 1 [( 6) .1 + ( 5) .1 + ( 4) .2 + .... + (5) .1 + (6) .1] = 8. 35 Therefore, 2 = 12 and S = 8.