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 ¼ the sum of the resulting numbers, and compute the pmf of w. then compute e(w) and var(w). [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 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.)

There are possible outcomes. At the least, you can draw (1, 2, 3) to get a sum of 6, and at most (5, 6, 7) for a sum of 18. Then the PMF is

That is, consider all the possible sums and their integer partitions. For example,

6 = 1 + 2 + 3

7 = 1 + 2 + 4

8 = 1 + 2 + 5 = 1 + 3 + 4

9 = 1 + 2 + 6 = 1 + 3 + 5 = 2 + 3 + 4

and so on.

Then the expected value is

We can find the variance via

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step-by-step explanation:

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